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ABSTRACT
Title of Dissertation: STUDENT SENSE-MAKING IN QUANTUM
MECHANICS: LESSONS TO TEACHERS
FROM STUDIES OF GROUP-WORK AND
REPRESENTATION USE
Erin Ronayne Sohr, Doctor of Philosophy, 2018
Dissertation directed by: Assistant Research Professor, Ayush Gupta,
Department of Physics
Associate Professor, Andrew Elby, Department
of Teaching and Learning, Policy and
Leadership
This dissertation covers two distinct threads of research; both threads focus on
understanding student-thinking in quantum mechanics and then draw implications for
future research and instruction. The primary goal of this collection of work is, in any
way possible, to improve instruction and find ways to better support students in their
learning.
The first thread of research focuses on tension negotiation in collaborative
group problem-solving. While group-work has become more commonplace in physics
classes, this research provides instructors some means of seeing just how complicated
group dynamics can be. In particular, I highlight one interactional pattern through
which students resolve tension emerging in group interaction by closing
conversations or conversational topics. In doing so, students leave some conceptual
line of reasoning unresolved. This work provides important insights into helping
instructors understand and respond to group dynamics and conversational closings.
The second thread of work focuses on flexible representation use. This thread
has two similar lines of research. The first focuses on how particular representations
(wavefunction and external potential graphs) associated with the infinite-well and
finite-well potentials can be used by students as tools to learn with. Adapting these
models to new situations can lead to deeper understandings of both the model being
adapted and the new situation. In some cases, the process of adaptation is not
impeded by the student lacking a sophisticated understanding of the model being
adapted.
The second line of research on representation use focuses on the reflexiveness
of student inquiry with representations. In reflexive reasoning, the student’s sense-
making shapes, and is shaped by, the representations they draw and animate. This
form of inquiry stands in contrast with traditional notions of proficiency in using
representations which tend to highlight reproducing standard representational forms
and then reading-out information from those forms. In this work, I highlight how this
non-linear, reflexive sense-making is supported by the development of coherent,
coupled systems of representations and attention to particular figural features, leading
to the generation of new meaning.
STUDENT SENSE-MAKING IN QUANTUM MECHANICS: LESSONS TO
TEACHERS FROM STUDIES OF GROUP-WORK AND REPRESENTATION
USE
by
Erin Ronayne Sohr
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park, in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2018
Advisory Committee:
Associate Professor Andrew Elby, Co-Chair
Assistant Research Professor Ayush Gupta, Co-Chair
Patricia Alexander
Kara Hoffman
Janet Walkoe
© Copyright by
Erin Ronayne Sohr
2018
ii
Dedication
To: Rosemary Grant Ronayne
iii
Acknowledgements
These will be brief.
UMD PER Group: Special thanks to my advisors Ayush and Andy, who have given
me far more of their time than I deserve. This work would not have been possible
without your guidance and the guidance of the rest of the group.
Friends and family: Thank you for all of your love and support over the years. You all
have kept me (mostly) sane through not only this program, but the twists and turns
that life has thrown my way.
My grandmother, Rosemary: I’m so grateful that I was able to care for you in the last
six months of your life. The amount of love and gratitude that you showed me each
and every day, often through tremendous pain, will never leave me. Your smiles and
kisses every morning when I woke you up and in the evenings when I put you to bed
keep me going through my darkest nights. In every way, I strive to be more like you.
The lessons I learned from you far outweigh anything else I have come to think I
know about the world. I know now to take care of the people and world around me,
just as you did. I pray for you (almost) every day. Some days I pray extra to make up
for the days I miss. But thankfully, because of the love you shared with the world,
many people continue to pray for you.
iv
To Joshie: There is so much I need to thank you for. Luckily, we have the rest of our
lives for me to express my gratitude. This has been a terrible year of loss and grief,
piled onto the stress of this program. As I write this, you are in the hospital, holding
our nephew as he says goodbye to his father. My heart breaks that I’m not there with
you two. Your unrelenting ability to support me and our families through these times
amazes me. So thank you for taking care of me, my grandmother, our nephew, our
friends, and our families in these times. I would not be here, writing this dissertation
or on this planet, if it weren’t for you.
v
Table of Contents
Dedication ..................................................................................................................... ii Acknowledgements ...................................................................................................... iii Table of Contents .......................................................................................................... v List of Tables ............................................................................................................. viii List of Figures .............................................................................................................. ix
Chapter 1: Introduction ................................................................................................. 1 References ................................................................................................................. 8
Chapter 2: Taking an Escape Hatch: Managing Tension in Group Discourse ........... 10 Abstract ................................................................................................................... 10
Introduction ............................................................................................................. 11 Literature Review.................................................................................................... 13
Tension in group-work: A view from management studies................................ 14
Resources for managing tension during group work in learning environments . 15 Situating our argument in this landscape ............................................................ 20
Methods................................................................................................................... 21 Data Context ....................................................................................................... 21 Background and Analytical Flow ....................................................................... 22
Methodological Orientation and Tools ............................................................... 24 Analysis................................................................................................................... 29
Episodes 1 and 2: “Because math” and “Can we define” ................................... 29 Episode 1: Because math .................................................................................... 30
Episode 2: “Can we define” ................................................................................ 47 Episode 3: Agreeing to disagree ......................................................................... 57
Discussion and Instructional Implications .............................................................. 69 Entanglement of cognitive and social tensions ................................................... 69 Escape hatches: a tension-relieving interactional achievement .......................... 70
Going meta on emotions ..................................................................................... 75 Conclusion .............................................................................................................. 77 References ............................................................................................................... 79
Chapter 3: Learning with and about toy models in QM ............................................. 88 Introduction ............................................................................................................. 88
Physicists adapt toy models to model more complicated quantum physics
scenarios .............................................................................................................. 92
Problematizing toy model adaptation in QM instruction .................................... 94 Adaptation as a form of learning ........................................................................ 95 Current physics education research has underexplored student reasoning in the
context of adapting toy models and iconic representations in QM to novel
situations ............................................................................................................. 97 Methods (Analytical Flow) ................................................................................... 101
How we created opportunities for adaptation of toy models and iconic
representations .................................................................................................. 101 Data Context ..................................................................................................... 102 Identifying Relevant Case studies for Analysis ................................................ 104
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Analytical Methods ........................................................................................... 105 Example Analysis: interview with Quinn on the classical well problem ......... 108
Results ................................................................................................................... 116 Preview ............................................................................................................. 116
Quinn on the classical well problem (continued) ............................................. 117 Infinite well representation as a first model for the bead-on-a-string ............... 117 Intervention: Attention to new regions shifts Quinn’s modeling of the bead-on-a-
string ................................................................................................................. 120 Reflections on the differences in ontology of the infinite well particle and the
bead-on-a-string ................................................................................................ 123 Discussion of Quinn .......................................................................................... 124 Oliver on the classical well problem ................................................................. 125
Discussion of Oliver ......................................................................................... 135 Chad on the slanted well problem ..................................................................... 136 Discussion of Chad ........................................................................................... 143
Conclusions ........................................................................................................... 143 Seeing continuities across situations ................................................................. 143
First instincts ..................................................................................................... 144 Coherence seeking across situations ................................................................. 145
Implications........................................................................................................... 146
Opportunities for adaptation ............................................................................. 146 What if adaptation leads to incorrect results? ................................................... 147
References ............................................................................................................. 149 Chapter 4: Designing Disruptions to Wavefunction Infrastructure for Flexible
Representation Use ................................................................................................... 158 Introduction ........................................................................................................... 158
Greeno and Hall’s invocation to teachers ............................................................. 160 Opportunities for Flexible Representation Use are needed in physics ................. 162
Current notions of reflexivity............................................................................ 162
Rigid Use of Representations can lead to disconnected Sensemaking ............. 165 How do we approach creating opportunities for flexible representation use?
Designing Disruptions to Representational Infrastructure .................................... 167 What is Representational Infrastructure? .......................................................... 169
Disruptions to Representational Infrastructure ................................................. 170 Designing for disruptions to representational infrastructure for generating
Wavefunctions ...................................................................................................... 178 Data Context ......................................................................................................... 179
Analytical Flow ..................................................................................................... 182 Identifying relevant cases ................................................................................. 182 Analytical Methods ........................................................................................... 183
Chad on the Classical Well Problem .................................................................... 189 Discussion of Chad on the Classical Problem ...................................................... 205 Chad on the Slanted Well Problem ....................................................................... 208 Discussion of Chad on the Slanted Well Problem ................................................ 219 Comparing the two episodes ................................................................................. 220 References ............................................................................................................. 225
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Chapter 5: Conclusions ............................................................................................ 230 Chapter 2: Tension in collaborative group-work .................................................. 230 Chapters 3 and 4: flexible representation use ....................................................... 233 Takeaways from interviews .................................................................................. 241
References ............................................................................................................. 243 Appendix 1 ................................................................................................................ 244
Transcript Conventions ......................................................................................... 244 Appendix 2 ................................................................................................................ 245
Tutorial on Doppler Cooling ................................................................................. 245
Instructor Guide: Tutorial on Doppler Cooling .................................................... 248 Appendix 3 ................................................................................................................ 251
Tutorial on Zeeman Effect .................................................................................... 251
Instructor Guide: Tutorial on Zeeman Effect ........................................................ 257 Appendix 4 ................................................................................................................ 264
Interview Protocol 1 .............................................................................................. 264
Interview Protocol 2 .............................................................................................. 269 Bibliography ............................................................................................................. 275
viii
List of Tables
2.1 Breakdown of types of contributions made to the discussion by Karen, Al, and
Larry…………………………………………………………………………………56
3.1 Example analysis of episode in interview with Quinn……………………...100
4.1 Example analysis of episode in interview with Paul……………………….180
ix
List of Figures 3.1 Quinn's Bead representation……………………………………………..…100
3.2 Quinn's Bead representation……………………………………………..…109
3.3 Quinn’s Probability Representation………………………………………...110
3.4 Quinn’s Refined probability Representation……………………………….113
3.5 Oliver’s Bead-on-a-string representation…………………………………..117
3.6 Oliver’s Probability Representation………………………………………..119
3.7 Oliver’s Velocity Representation…………………………………………...124
3.8 Oliver’s Refined Probability and velocity Representations………………...126
3.9 Chad’s Slanted well representation…………………………………………128
3.1 Chad’s Finite well representation…………………………………………..128
3.11 Chad’s Slanted well representation………………………………………...130
3.12 Chad’s Infinite well representation above the slanted potential
representation……………………………………………………………………….132
3.13 Chad’s Slanted well representation…………………………………………133
4.1 Naval nomograph…………………………………………………………...175
4.2 Chad's Probability Representation………………………………………….182
4.3 Chad's Position Representation……………………………………………..184
4.4 Chad's Revised Probability Representation………………………………...195
4.5 Chad’s Slanted Well Representation……………………………………….202
4.6 Chad’s Finite Well Representation…………………………………………202
4.7 Chad’s Infinite well representation above the slanted potential
representation……………………………………………………………….............205
1
Chapter 1: Introduction
This dissertation is a collection of work developed during my time working
within the community of Physics Education Research. Work done in this community
is normally spearheaded by physicists and aims to understand the teaching and
learning of physics (McDermott & Redish, 1999). Broadly, the ultimate goal of work
within this field is then improving physics instruction (McDermott, 2001). My work
shares this orientation. On a large scale, there are three ways in which I contribute to
this goal: 1) developing curricular materials for physics classrooms, 2) studying
factors that influence students’ reasoning and learning, and 3) understanding what
lessons can be abstracted for instructors to then help support students. This
dissertation focuses on the latter two of these sub-goals; studying student reasoning
and drawing insights for instructors from these studies.
In this introductory chapter, I’ll briefly touch on each of these.
Curriculum development in quantum mechanics sets the context of my research
studying student thinking
My work in curriculum development helps set the context for my dissertation
work. In particular, I have worked on two curriculum development projects for
undergraduate quantum mechanics courses. The first project focused on engaging
students in reasoning about ontology; ideas about the properties of quantum objects
2
(i.e. particle or wave) and how those entities therefore interact with the world. The
second project has focused on developing materials that support mathematical sense-
making; different habits of mind in which students see coherence between
mathematical and physical structure.
Examples of two tutorials I have taken the lead in designing can be found in
the appendices of this manuscript. They both focus on laser-cooling of atoms. The
instructional goal of these tutorials is to give students opportunities to think about
more “real-world” situations than they might normally find in introductory quantum
mechanics courses. Granted, by real-world I mean laboratory-generated. In any case,
the tutorials allow students to apply their quantum formalism to understand how
physicists go about studying many-body quantum behavior.
Curriculum development in quantum mechanics provides means of data
collection for dissertation work
My work in curriculum development also provided a setting and pattern of
data collection for the rest of my work. The first data that I collected and studied for
these projects came from testing out tutorials in small collaborative groups of
students. Trying to understand the efficacy of different tutorial prompts through these
focus groups provided the data for my first body chapter. Data collection for the 2nd
and 3rd body chapters was taken in a similar vein; problem-solving interviews with
engineering and physics students. The data from these chapters were more ‘targeted’
in the sense that the problems given to students had a specific focus on
3
representation-use. The data collected from these interviews maintained a focus on
understanding how students generally respond to different opportunities for problem-
solving.
Through this process, I eventually amassed a collection of videotapes of
students working on tutorial-style problems. These problems are often fairly ‘short’ in
nature, in that a given tutorial may contain on the order of a dozen questions. These
problems are often conceptual in nature, tend to not involve sufficient calculation, and
aim to provide a good basis for discussion among the students. In my experience,
there can be a wide range in the amount of time it takes an individual student, or a
group of students, to get through a single question. The amount of time may range
from less than a minute to 30minutes. From consideration of the entire data
collection, students typically spend an average of five minutes on each problem.
My research involved studying these relatively short moments of students’
reasoning and distilling implications for researchers and instructors. It may seem
strange that a dissertation that focuses on learning and lessons for physics instructors
would focus on student reasoning that spans only a few minutes in duration.
However, in the next section, I briefly explain why I’m interested in such moments.
More motivation for studying different aspects of sense-making can be found in the
main chapters of this dissertation.
Studying student sense-making and learning: focusing on moments of
reorganization
4
I see sense-making and learning as the coordination within a complex system
of interaction (Newman, 2011; Hammer, Elby, Scherr, & Redish, 2005; Hutchins,
1995; Greeno, 1998). The collective behavior of the system arises from interaction
between its constituent parts, those parts being people and their material
surroundings. In this perspective, moments of reorganization or coordination become
crucially important. This is because learning occurs when patterns of reorganization
become internalized by individuals within the system (Hutchins, 1995). The question
then becomes how students move from spending a few minutes making connections
about a topic to deep, meaningful learning.
To answer this question, I will first discuss characteristics of sense-making
that I see as valuable in reasoning about physical situations. Arguably, the main goal
of physics is to develop coherent explanations and models to explain physical
phenomena. And so sense-making that becomes particularly important here are forms
of mechanistic reasoning. According to Russ, Scherr, Hammer, and Mikeska,
mechanistic reasoning should involve any/all of the following: describing a target
phenomenon, identifying set-up conditions, identifying entities, identifying activities,
identifying properties of entities, and identifying the organization of entities.
Connections among these features may discursively appear in the forms of chaining,
analogies, or animated models (Russ, Scherr, Hammer, & Mikeska, 2008).
For example, consider a student reasoning about the probabilistic behavior of
a bouncing ball. The student then makes an analogy, comparing the bouncing ball to a
quantum particle. The analogy allows the student to reason about how the physical
5
differences between the bouncing ball and the quantum particle gives rise to different
probabilistic behaviors. Here, the student is sense-making through making
connections about the entities involved (types, attributes, and activities) through an
analogy to develop a deeper understanding of the classical “particle-in-a-box”1.
This research is set-up to study short patterns of sense-making across multiple
groups of students. This work is not geared towards studying long-term learning by
students. Instead, when I talk about learning, particularly in chapters 3 and 4, I’m
more specifically considering what Schwartz, Bransford, and Sears (2005) would call
preparation for future learning. Preparation for future learning concerns ways in
which students are preparing themselves to learn further about a topic. Such “seeds of
learning” may be seen in growth in verbalizations about a topic, questions being
asked, resources used or requested, or redirections in perceptual attention. Schwartz
and Martin suggest that this type of student invention helps students notice
distinctions or important features that then guide their future learning, (Schwartz and
Martin, 2004).
My research, particularly chapters 3 and 4, typically looks for preparation for
future learning in situations where students often do not have the requisite knowledge
needed to simply replicate or directly apply what they already know. Instead, students
must be adaptive and inventive in their sense-making. In these situations, the
reasoning that students are doing may sometimes look non-canonical. However,
1 See Chapter 3 for more detail.
6
previous work has shown that opportunities for invention early on can lead to better
learning gains than providing students opportunities to participate in tell-and-practice
methods, (Schwartz and Bransford, 1998; Schwartz and Martin, 2004). For example,
Schwartz and Martin found that students who invented methods for statistical
comparison were then better able to learn from a worked example than students who
were initially given, and then practiced, the canonical method.
Two threads of research emerged in studying student reasoning
Chapter 2: Tension in collaborative group problem-solving
As mentioned earlier, a primary goal in watching focus group data is to
understand how students respond to the tutorials written by my research group. The
goal being that student responses to the tutorial prompts can help inform revisions of
those prompts. In these viewings, an interesting episode stuck-out. In a bout of
particularly tense sense-making, a group of students came to play on the wording of
the tutorial to find an ‘out’ from their tense conversation. Moving between viewing
the data collection and collective discussions with the research team, the pattern held.
Students were finding creative ways to find ‘outs’ from tense episodes of sense-
making. This work seeks to better understand this pattern of reasoning, which we call
‘taking an escape hatch’. In particular, this work focuses on how tension can play a
driving role in conceptual sense-making.
7
Chapters 3 and 4: Representation-use in individual student interviews
These chapters focus on student reasoning with representations. Unlike the
first study on tension in group problem-solving, the data from these chapters come
from individual student interviews. There were two interview protocols used to
collect the date for these chapters. These protocols can be found in the appendices at
the end of this dissertation.
In proctoring and reviewing these interviews, I saw that the interview space
and prompts allowed particular types of ‘representational play.’ Students drew
representations and pictures in their reasoning and would proceed to break them
apart, manipulate them, piece them back together, etc. These acts were highly non-
canonical and very creative. It seemed further that these actions were generative
towards the student’s endeavor of developing deeper understandings of the situations
I posed to them. I found these actions to stand in stark contrast to more traditional
notions of representation-use, where a representation simply reflects a student’s
thinking or makes it easier to perform simple manipulations or read-outs.
These two chapters are both geared towards understanding this type of
‘flexible representation use,’ which will be defined more thoroughly in Chapter 4.
Though similar, I have chosen to split the work into two separate chapters in order to
help focus instructor attention on different aspects of student sense-making. From
these chapters I draw complementary instructional implications.
8
In the concluding chapter of this dissertation, I reflect more on the decision to
separate the two lines of work on representation use. I also touch more on the
instructional ‘lessons learned’ from this work. Many of these takeaways have to do
with helping develop instructional practices of noticing. I.e. As an instructor, what are
the things I should attend too and how? Additional implications about task design are
also discussed.
References
Greeno, J. G. (1998). The situativity of knowing, learning, and research. American
psychologist, 53(1), 5.
Hammer, D., Elby, A., Scherr, R. E., & Redish, E. F. (2005). Resources, framing, and
transfer. Transfer of learning from a modern multidisciplinary perspective, 89.
Hutchins, E. (1995). Cognition in the Wild. MIT press.
McDermott, L. C. (2001). Oersted medal lecture 2001:“Physics Education
Research—the key to student learning”. American Journal of Physics, 69(11),
1127-1137.
9
McDermott, L. C., & Redish, E. F. (1999). Resource letter: PER-1: Physics education
research. American journal of physics, 67(9), 755-767
Newman, M. E. J. (2011). Resource letter cs–1: Complex systems. American Journal
of Physics, 79(8), 800-810.
Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and
instruction, 16(4), 475-5223.
Schwartz, D. L., Bransford, J. D., & Sears, D. (2005). Efficiency and innovation in
transfer. Transfer of learning from a modern multidisciplinary perspective, 1-
51.
Schwartz, D. L., & Martin, T. (2004). Inventing to prepare for future learning: The
hidden efficiency of encouraging original student production in statistics
instruction. Cognition and Instruction, 22(2), 129-184.
10
Chapter 2: Taking an Escape Hatch: Managing Tension in
Group Discourse
Abstract
Problem solving in groups can be rich with tension for students. This tension
may arise from conflicting approaches (conceptual and/or epistemological), and/or
from conflict emerging in the social relations among group members. Drawing on
video records of undergraduate students working collaboratively on physics
worksheets in groups of 4-5, we use three cases to illustrate the multifaceted ways in
which conflict arises—combining conceptual, epistemological, emotional, and social
dynamics—and a specific way of managing the tension that can emerge from the
multifaceted conflict, that we call “taking an escape hatch.” An escape hatch is a set
of discourse moves through which participants close the conversational topic, thereby
relieving tension, but before a conceptual resolution is achieved. We describe how
epistemological twists and turns can be recruited as a means of managing the strong
emotions experienced by the students, showing the coupling of emotion and
epistemology in students’ conceptual sense-making during group-work. In doing so,
we help to provide the groundwork necessary for instructors to notice, understand,
and respond to one way in which conceptual-epistemological—social-emotional
aspects of interaction are coupled in the emergence of tension, rather than narrowly
targeting instructional moves based on only conceptual or epistemological
considerations. Instead, instructors should often respond to— and help students
11
become aware of—the emotional component of peer interactions and its entanglement
with the “cold cognitive” conceptual and epistemological components.
Introduction
Collaborative, active learning in small group settings using research-based
materials can have many benefits (Alexopoulou & Driver, 1996; Barron, 2000;
Heller, Keith, & Scott, 1992). Group learning allows students to share knowledge as
they build on and critique each other’s ideas and reasoning strategies. This creates the
opportunity for students to participate in better problem-solving approaches and
solutions than when working individually (Heller, 1992). However, collaborative
problem solving can create challenges due to the necessarily social negotiation of
ideas, approaches, and communication styles (De Dreu & Weingart, 2003; Johnson
and Johnson, 1979). Collaborative learning can give rise to conflict for a variety of
reasons. The ideas introduced in the group and their connection to the end goal may
be unclear. What is taken to be understood by the group can fluctuate quickly.
Because common ground is so variable, it demands constant attention by participants
(Barron, 2000). Conflicts may arise from dominant personalities (Heller, 1992),
unequal opportunities to participate (Sullivan & Wilson, 2013), failure to obey turn-
taking norms, or students’ insistence on their own strategies (Barron, 2000). Previous
research has shown students using epistemic distancing (proposing ideas without
taking ownership of them; (Conlin, 2012) or slipping into less collaborative modes
(Barron, 2003) as ways of mitigating the tension that arises in the face of these very
12
different types of conflict. A better understanding of the nature of group conflicts,
including the interactional processes underlying their generation, sustenance, and
resolutions, can help both designers and facilitators of small-group learning activities.
This is particularly true for conflict that gives rise to emotional tension.
In this paper, we study different analytical dimensions of interaction
(conceptual, epistemological, social, and emotional) and their interaction. In
particular, we highlight how the emotional/affective analytical dimension of
interaction is entangled with the other dimensions. In doing so, we (i) contribute to
the small body of work focusing on the entanglement of conceptual, epistemological,
emotional, and social dynamics in small group work, and (ii) characterize a type of
student interaction during tense group negotiations, which we call “escape hatches.”
By “escape hatches” we mean collaboratively achieved closings of tense discussions
leaving unresolved the core conceptual issue(s) that formed the context of the local
conflict. To do so, we analyze three episodes of students engaged in collaborative
physics problem-solving, showing how tension arises in the emergence of
multifaceted conflict. In doing so, we document a variety of conversational moves
that can initiate students’ taking an escape hatch, thereby relieving tension.
We show, in one case, that escape hatches can emerge as epistemological
stances or humor. This complicates the facilitator’s job, as the nominal meaning and
discursive function of students’ utterances can differ radically in hard-to-notice ways.
We start by reviewing literature on conflict and tension in teamwork, in both
professional and in educational settings. Then we outline the data collection and
13
analytical flow of our work. Next, we present our analysis of three episodes of small
group work showing how the escape hatch the students’ take function in the groups’
discourse. We conclude with implications for research and instruction.
Literature Review
Collaborative problem-solving groups have long been seen as helping people
learn complex skills (Collins, Brown, Newman, 1989; (Brown & Palincsar, 2013;
Heller et al., 1992). Still, when people work together, disagreements often arise. The
literature on argumentation and conflict in collaborative work (Bricker & Bell, 2008;
Mortimer and Machado, 2008; Kutnick, 1990; Lawson, 1995; Berland & Reiser,
2011; Aikenhead, 1985) often focuses on the conceptual and epistemological aspects
of group work. Few studies simultaneously attend to the emotional and relational
aspects of group work, and even fewer simultaneously attend to those aspects and the
conceptual and epistemological aspects. In this brief walk through the literature, we
focus on the latter studies to document that (i) some studies suggest that the
conceptual, epistemological, and social aspects of conflict during group-work are
coupled and (ii) students have a variety of tools to manage the tension that can arise
with conflict in an interaction. For (i), we draw on management studies of conflict in
professional settings before turning to education research, which typically addresses
the two points simultaneously. We close this section by arguing that fine timescale
investigations of how these conflicts arise and are resolved are still needed,
motivating this paper.
14
Tension in group-work: A view from management studies
Within organizational and management studies, researchers have classified
conflict as affective/interpersonal or as cognitive, with cognitive conflict arising from
different conceptualizations of the task and from disagreements about resource and
process management (K. A. Jehn & Mannix, 2001) Shah & Jehn, 1993; (Amason &
Sapienza, 1997; De Dreu & Weingart, 2003; K. a Jehn, 1997). These studies
document complicated patterns of coupling between conflict and team performance,
influenced by the interactions between different conflict types, task types, and team
dynamics. Overall, teams experiencing less conflict (affective and cognitive) tend to
perform better (De Dreu & Weingart, 2003; K. A. Jehn & Mannix, 2001). Yet, Jehn
and Mannix (2001) found that more successful groups tended to experience rising
conflict over time, while some lower-performing groups experienced a dip in task
conflict halfway through. In contrast, De Dreu and Weingart (2003) found that
conflicts are less disruptive for simpler and shorter-term tasks than for complex, long-
term projects. And, task conflict has a lower impact on performance when task and
relationship conflicts are weakly correlated. They argue that “teams benefit from task
conflict when they cultivate an environment that is open and tolerant of diverse
viewpoints and work with cooperative norms preventing those disagreements from
being misinterpreted as personal attacks (Amason, 1996; De Dreu & West, 2001; K.
Jehn, 1995; Lovelace, Shapiro, & Wiengart, 2001; Simons & Peterson, 2000).
These findings in organizational/management studies have implications for
research in science learning. For one, even in science problem-solving tasks, we
15
should expect that cognitive conflicts (resulting from conceptual and/or
epistemological differences) might entangle with affective or relational conflict. This
entanglement is still underexplored in science education, where most studies have
focused on cognitive or interpersonal/affective conflict. Another implication is
methodological. The finding that a group’s performance depends not simply on the
amount but on the types and timing of the conflicts suggests that, in studying small-
group learning and problem-solving in science, we will obtain incomplete or even
misleading results if we look only at coarse-grained relations between “level of
conflict” and “performance.” We need fine timescale examination of conflict arising
during group work, including their genesis and resolution. This is precisely the charge
this paper takes on.
Resources for managing tension during group work in learning environments
In this section we present illustrative episodes from the few studies in science
and mathematics education literature that explore possible entanglement between the
cognitive and affective/relational aspects of group-work, to provide a feel for and to
situate our argument within previous work.
Lampert et al. (1996) discuss fifth graders’ actions in the face of disagreement
while working on a math problem concerning a car traveling at constant speed. A
group of four students in the back of the room, “talking loudly and gesturing toward
one another” (p. 748), stands out to Lampert (the teacher/researcher). Within this
group, a disagreement occurs when Sam misreads Connie’s answer, mistaking “min”
16
to denote miles. While the group discusses whether the answer should be in minutes
or miles, Sam “seems to be trying to reduce [Connie’s] credibility with the others in
the group, especially when he accuses Connie of ‘guessing’ rather than ‘figuring it
out.’” Thus, the conflict in this situation is simultaneously characterized by a
conceptual layer (minutes or miles), an epistemological layer (guessing versus
figuring it out), and an interactional positioning layer (establishing relative status).
The group settles on “minutes” and moves on to another disagreement over the
correct numerical solution. Again, Connie and Sam are at odds, with Connie
supplying the correct answer in response to Sam’s incorrect solution. Sam and Connie
go back and forth trying to persuade the other group members, Enoyat and Catherine.
Sam and Connie then implicitly agree to disagree with Sam noting “I’m just putting 1
hour 20,” and Connie noting that “I’ll put 1 hour 40.” This move confuses the other
group members, who conceptualize the mathematical activity as including coming to
consensus. Enoyat is unsure of how to accomplish this task. However, he proposes
that he average the two responses in an “attempt to resolve the discomfort he feels in
choosing between Connie and Sam” (pp. 751).
According to Lampert, Sam’s action initially supported the belief that the
mathematical discussion should include coming to consensus, but later “his
mathematical intention also gets confounded with a social one as he seems satisfied
with everyone ‘writing what you think the answer would be.’” (pp. 754) Specifically,
in this conflict, the group begins with a conceptual negotiation (over units, then
numbers), which gives rise to both conceptual and interpersonal conflict. However,
17
the group manages these conflicts at the boundaries of the social and epistemological
layers; they renegotiate the rules about what counts as a valid answer, with consensus
no longer a criterion. For Enoyat, the situation involved an emotional layer as well, in
that his resolution aimed to resolving the tension that was associated with the
epistemological and social conflict being created by Sam and Connie. So, as Lampert
emphasizes, the “joint activity [of generating tension and resolution] is not just an
expression of what they bring to this conversation by way of beliefs about how to
disagree—they are shaping those beliefs dynamically as they interact” (Lampert,
1996, p. 754). Hence, “reasoning and social negotiation become intermingled. In a
mélange of social and mathematical moves, the students struggle to figure out how to
both maintain their relationships and do what the teacher has asked.” (pp. 751). In
summary, the epistemological, social, emotional components of conflict were coupled
and negotiated in the moment.
Taking another tack, a few researchers have looked at humor, playful talk, and
skillful positioning of ideas as ways to navigate conflict in group-work. Conlin (2012)
shows how students use humor and irony to manage the affective risk of threatening
face (Goffman, 1955) when making repairs to each other’s conceptual reasoning.
Students also manage the threat to face through “epistemic distancing”, a shift of
footing (Goffman, 1955) wherein the student positions herself as the messenger of
someone else’s claim rather than the claim’s author. If the claim is rejected or
repaired, the loss of face is therefore shifted away from the messenger.
Methodologically, Conlin found that epistemic distancing can be evident not just in
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the substance of a student’s utterance but also through other “paralinguistic channels,
such as shifts in register and prosody, facial expressions and gestures” (Conlin, 2012;
Goodwin, 2007).
Similarly, Sullivan and Wilson (2015) document young students’ use of
playful talk (humor, puns, teasing, music making, and wordplay) (Lytra, 2009;
Sullivan & Wilson, 2013) as a means navigating conflict in small group work. In their
case study of 6th grade science students building a robot, conflict arises with respect
to status within group and associated access to the work of the project (who gets to
build the robot, whose ideas are taken up), perceived gender identities, and other
flashpoints. The students used playful talk to manipulate opportunities to participate
within the group by positioning themselves or others as more or less capable. For
example, one of the group members was often positioned as less competent by her
peers. When the group was allocating building tasks, she playfully offered to build
the entire device, positioning herself as a competent builder within her group and
staking out a slot on the building team. So again, in this study, cognitive conflicts
(e.g., over what ideas get taken up) are entangled with social conflicts (e.g., over who
gets to participate in what ways).
Barron (2000, 2003) argues that conflicts during group work can arise because
students bring different orientations towards what it means to collaborate. Some
students behave in ways that support equitable participation and joint attention to
ideas and artifacts. They align their task-orientation through referencing and building
on one another’s ideas and approaches. Others want to dominate the discussion,
19
insisting on control and authority. Note that these two different orientations have
both an epistemological component (is knowledge collaboratively constructed or
authoritatively transmitted?) and a social component (more equitable vs. less
equitable participation patterns). When group members consistently approach
collaboration in these different ways, conflicts arise and performance can suffer.
Different orientations towards collaboration are made visible through “struggles of
control, failures to understand one another, repeated attempts at explanation,
rejections of that explanation (even when invited), self-focused talk, admissions of
confusion,” (Barron, 2003, p. 366), etc. Barron argued that students try to manage and
negotiate their forms of participation, and expectations thereof, during group-work.
And whether initial differences in participation converge or further diverge depends
both on social and cognitive factors.
How people organize their participation in conversations and the generation
and resolution of conflict is also an area of study within sociolinguistics. Goodwin
(2007), for example, describes the organization of embodied participation
frameworks in an episode in which a father is attempting to help his daughter with her
homework. Embodied participation frameworks concern the embodied alignment and
organization for talk and action within an interaction. Like the students in Barron’s
(2000) paper, the father and daughter also brought different expectations of
collaboration to their interaction, as made visible through their talk (substance of
utterances as well as tone, pitch, etc.), body posture, gestures, and gaze. While the
father wants to help the daughter figure out the homework, the daughter wants him to
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just tell her the answer. This conflict leads to a breakdown and collaboration could
only resume once their participation frameworks were better aligned.
Situating our argument in this landscape
This brief walk through prior work suggests that (i) for researchers,
understanding conflict arising in group work requires the simultaneous attention to
social-interactional aspects and the cognitive aspects of the interaction, and (ii) for
students, a resolution of tension that enables the collaborative work to proceed
smoothly often requires alignment along some of the cognitive and/or social
dimensions. This manuscript both builds and expands upon this prior work, by
(i) illustrating that in the genesis and sustenance of group tension, the cognitive,
affective, and social components are not only simultaneously present but dynamically
coupled, mutually affecting each other; and
(ii) introducing the notion of “taking an escape hatch” as one way in which groups
relieve tension.
In taking an escape hatch, the group relieves the affective and social conflict
but without resolving the cognitive disagreements that helped produce those conflicts.
Lampert et al.’s documentation of “agreeing to disagree” is an example: By
renegotiating what counts as an acceptable answer (deciding that consensus isn’t
necessary), the group closes the tense conceptual discussion about how much time the
21
car takes, thereby relieving the social conflict generated by Sam and Connie. The
episodes we present below suggest that “agreeing to disagree” is just one of many
ways of taking an escape hatch, and that “taking an escape hatch” may be common in
students’ collaborative small-group work in science.
This paper also contributes to the need for more empirical analyses that bridge
cognitivist and interactionist analysis (diSessa, Sherin, Levin, 2015). In addition, little
research provides fine timescale analyses of discourse in undergraduate-level
collaborative learning, especially in upper-division disciplinary contexts such as
quantum mechanics.
Methods
Data Context
As part of our design process in creating curriculum materials for upper
division quantum mechanics courses, we video recorded groups of 3-5 students
engaging with the materials developed. The curricular materials were in the form of
worksheets which posed sequences of conceptual questions, to be answered
collaboratively by the group. Physics and engineering students (mostly juniors and
seniors), were recruited for these groups through an email to the first semester of the
physics-major quantum mechanics class or through a department-wide email.
Focus group sessions were held in a room in the physics building. Each
session was attended by 3-5 students and the interviewer. These sessions started with
22
the interviewer explaining to the students that the researchers were interested in how
the students responded to the tutorials and how the students tend to think and talk
about quantum mechanics, more generally. The groups of students typically
proceeded through the tutorial with very minimal, unprompted input from the
interviewer.
In some cases, the same group of students attended multiple sessions. Because
of this, the students were able to get to know the researchers (myself included), the
other students (if they did not already know them), and the norms associated with the
focus group space. Each episode presented below will include some detail about the
students in groups, including the nature of their participation in focus groups. In the
discussion section, I will make some conjectures about how differences in
relationships with these students may have applicability for the findings of this paper.
Background and Analytical Flow
In studying the data collection, we were broadly interested in students’
reactions to the worksheets, moments of struggle (conceptual or otherwise), moments
of negotiation and coordination among students, and the role of ontologies (Brookes
& Etkina, 2007) and metacognition in students’ reasoning. We worked inductively
and deductively (Erickson, 2006), viewing the data on a larger scale and then
selectively investigating areas of interest more closely (Derry et al., 2010; Jordan &
Henderson, 1995).
23
Specifically, the first author began by watching the data with an inductive
orientation, looking for patterns in the data, as guided by emergent interest and
commitment to attending to fine time scale variations in students’ talk and interaction.
When viewing a group of students working through our Particle in a Box worksheet,
the first author noticed an interactional pattern that occurred twice during the session.
During these episodes the discussion became quite tense, as evidenced through
volume of speech, patterns of cutting other speakers off, and body language. At this
stage of analysis, the first author was attending to these indications more intuitively
rather than following any strict methodology or pursuing a specific research question.
In the midst of these two tense moments, which both occurred during the throes of
group problem-solving, students escaped the tension by making and taking up a bid to
close the conversation or topic (Schegloff & Sacks, 1973), but without coming to a
conceptual resolution.
The first author then brought the video episodes and transcripts to video
analysis sessions (Jordan & Henderson, 1995) attended by all the authors and
sometimes by collaborators from another university as well. The transcripts at this
stage did not yet include intonation, stresses, or gestures; for those, we relied on the
video. As a group, we formed alternative interpretations of the data and tested those
via repeated viewings in which we would expand on the layers of multi-modal
analysis (Stivers & Sidnell, 1998) to see which interpretations were best supported by
coherence across multiple channels of talk and action. We labeled the interactional
pattern “taking an escape hatch,” and worked collaboratively to characterize the
24
mechanics of this type of interaction. Further nuance to the phenomenon developed
through reflexively moving between the data and operationalization of the
phenomenon. Ultimately, this process lead us modeling the interaction of “taking an
escape hatch” as having three main, coupled characteristics.
1) The move functions to relieve tension within the group.
2) The move closes discussion of the current conceptual and/or
epistemological topic of the conversation.
3) Taking an escape hatch circumvents finding a conceptual resolution for
that topic.
Within these constraints, a variety of conversational moves can function as bids for
taking an escape hatch, as we document in this paper.
Methodological Orientation and Tools
We are examining the emergent intertwining of the conceptual, social and
epistemological dimensions of group interactions around physics problem-solving. In
order to make empirical claims to whether or not a move constitutes an escape hatch,
we utilize talk-in-interaction as a primary data source (Derry et al., 2010; Goodwin,
2007; Jordan & Henderson, 1995), analyzing multimodal semiotic channels (speech,
gesture, material ecologies) to develop a coherent story of a group’s collaborative
interaction, which can then be binned into more conceptual, affective, and
epistemological layers. Because we want to understand the process and mechanisms
comprising and supporting the taking of an escape hatch in group interaction,
25
microgenetic analytical methods provide an empirical framing for doing so (Parnafes
& diSessa, 2013; Siegler & Crowley, 1991).
Knowledge-in-use analysis to attribute conceptual substance. In unpacking
the conceptual substance of students’ talk, we attend to fine shades of meaning (e.g.,
the same word taking on different meanings at different moments) and to the changes
in conceptual meaning that happen at short timescales. We don’t assume coherence of
“conceptions” across or within students unless warranted by features of their talk and
action. Work that exemplifies analysis of conceptual knowledge-in-use comes from
Beth Warren, Ann Rosebery, and colleagues (Warren, Ogonowski, Pottier, 2005;
Rosebery & Puttick, 1998). Like Rosebery and Warren, we loosely draw on knowledge
analysis (DiSessa,1993; Hammer, 2000) without aiming to model the knowledge being
enacted in terms of cognitive elements or making claims about the ontology of
knowledge-in-use.
Epistemological statements and strategies to attribute epistemological
substance. To understand the role of epistemology in students’ interactions, we
attend to students’ explicitly stated stances towards knowing in the moment, as well
as their tone, hedge words, disclaimers, organization of available material resources,
and coordination of their activity in order to produce knowledge. Students’
negotiations around what counts as a satisfactory answer and how to approach a
problem provides strong empirical characterization of how knowing and learning are
being enacted in the moment. For example, students approaching a problem and only
discussing formulaic or highly mathematized ideas are functionally approaching the
26
problem as having a mathematical solution path. This methodology is consistent with
Goodwin’s (2007) analysis of epistemic stances. Thus our analysis takes a “social
practices” rather than a “beliefs” perspective towards epistemology (Kelly,
McDonald, & Wickman, 2012).
Interaction analysis to attribute smooth vs. tense interaction. Tools from
interaction and conversation analysis provide a means for understanding the micro-
scale organization of talk-in-interaction. We describe interaction in through the
following dimensions/structures:
● Turn-taking; individual turns at conversation
● Repair; attempts to alleviate conversational trouble or breakdowns in mutual
understanding
● Turn construction; conversational turns are structurally comprised of turn-
construction units, which may be single words, clauses, questions, etc.
● Adjacency pairs (Sidnell, 2010; Sacks and Schegloff, 1973); distributed
conversational sequence of two utterances, where the first-pair part mutually
constrains second-pair part
● Preference; some second-pair parts are organizationally “preferred” in the
sense that some second-pair parts make more significant progress towards the
joint enterprise underway
27
● Progression; on a larger scale than preference, there is a sense that the
conversation should move towards accomplishing the mutually determined
purpose
We also rely on paralinguistic features of speech, such as tone, volume, and pauses.
Taken together, these tools help us understand what group members are (more or less)
jointly trying to accomplish, how they are going about accomplishing these actions,
what resources (knowledge, skills, experiences, etc.) the group utilizes in doing so,
and what emotions are evident through physical presentation. This analysis shows
when joint action unfolds smoothly, or when there is some conflict within or across
any of the aspects of action described above. A paradigmatic example we draw upon
is Goodwin’s (2007) analysis of a father helping her daughter with math homework;
the father initially approached the interaction as helping his daughter figure out the
answers, while the daughter initially wanted her father to simply provide the answers.
Goodwin used the substance of utterances as well as tone, pitch, body posture,
gestures, and gaze to document tension arising from different orientations toward the
interaction. Evidence of this tension comes from measures of embodied opposition
such as vowel lengthening, volume, gesture, posture, pauses and polarity markers at
the outset of conversational turns (Goodwin & Goodwin, 2002).
Attribution and categorization of conflict. We then sought to model the
conflict present in each interaction, which we characterized as social, conceptual, or
epistemological. Categorizing conceptual, social, or epistemological conflict involved
28
describing any extended opposition/decoherence in: how people are relationally
involved in the interaction (social), how knowledge is being enacted or constructed
(epistemological), and the content of the interaction (conceptual). For example, a lack
of conceptual progression, with students positioning different conceptual ideas or
approaches against each other, would be evidence only of conceptual conflict. We
might also expect these types of conflicts to be correlated in their emergence. For
example, a conversation in which there are a high degree of cut-offs and
interruptions, particularly of one person is an example of social conflict, where there
is a large degree of opposition, disjointness with respect to how people are
relationally related. When this interactional pattern comes to affect how knowledge is
being enacted or constructed, the conflict has clearly taken on both social and
epistemological dimensions.
Adjacency pairs to attribute conversational closings. We now turn to the
mechanics of the discursive moves that typically constitute an escape hatch. (From
here forward, we use “escape hatch” as shorthand for “taking an escape hatch.”)
Because an escape hatch is a way to close a conversation in response to tension, we
utilize Schegloff and Sacks’ formulation of adjacency pairs as a prevalent means of
identifying conversational closings. Adjacency pairs, also called “possible pre-
closings,” are particular examples of a two-part sequence of conversational turn-
taking in which the first utterance, a bid to close the discussion, constrains the second
utterance. (Schegloff & Sacks, 1973). For example, a student may make a bid to close
with “alright?” which when met with “alright” is an agreement to close, but when met
29
with “wait, what happens tomorrow?” is rejection of the bid. Thus, closings are
interactionally distributed.
Analysis
In the following sections, we present analysis of three episodes of students
working in extra-curricular focus groups on worksheets of quantum mechanics
problems.
The first two episodes come from a focus group that took place in late 2014.
Five students participated in the group; they were all male, and junior or senior
physics majors. They knew each other, to varying degrees from the quantum class
they were currently enrolled in, and other common courses. Their pseudonyms are Al,
Bob, Chad, Dan and Ed. Approximately four months later, Al and Ed returned for a
second focus group session. This time, they were joined by Karen and Larry, also
upper-level physics majors. The third episode in this paper comes from this second
focus group session. We named the episodes “Because math,” “Can we define” and
“Reframing” based on the content of student interaction during the episodes.
Episodes 1 and 2: “Because math” and “Can we define”
The first two episodes of escape hatch that we present, occurred in the clinical
focus group session using the worksheet on the Particle in a Box (PIAB)2. The PIAB
worksheet has students consider the properties of the quantum particle bound within a
2 https://www.physport.org/curricula/QuantumEntities/
30
square potential well, a standing wave on a string (as an analogy to the energy
eigenstates), and a classical particle in a box. Episodes 1 and 2 occurred about
5minutes and 15minutes into the hour-long focus group, respectively.
In “Because math,” the students are considering the question:
Why isn’t the ground state n = 0? That is, why isn’t it possible for the
particle to have zero energy?
In this episode, the students engaged in a tense discussion of the mathematics of
eigenstates and eigenvalues before dissipating that tension via terminating that line of
reasoning. In the second episode, students are discussing the question;
Can we define a ‘speed’ for the wave?
This question references a classical standing wave on a string. Here, the students
engage in a long period of tense reasoning before defusing tension by taking an
escape hatch afforded by the wording of the question.
Episode 1: Because math
After Chad begins reading the question out loud, Al suggests a conceptual
solution, but it isn’t taken up. Al then suggests a more mathematical path forward.
(Transcript conventions used are presented in the Appendix I.)
31
Segment 1/6: Lack of input influences framing
There is a three second pause after Al suggests the “uncertainty principle”.
Chad looks up from his paper and frowns as he looks to Al but he, and the rest of
group, remain silent. Al amends his suggestion with a hedge: “I guess
mathematically, I don’t know why.” indicating that the group’s silence and Chad’s re-
focusing his attention are taken up by Al as the group not taking up his suggestion.
This amendment has epistemological connotations, by suggesting that “know[ing]
why” may involve thinking “mathematically.” In the subsequent conversation, the
group takes up the bid to pursue a mathematical explanation, exploring entry points
while drawing heavily on mathematical language.
Segment 2/6: Grappling for an entry point in a mathematical space
32
33
Here, although the students are on the same page about taking a mathematical
approach (as indicated by repeated mentions of the energy equation for a harmonic
oscillator vs. a square well), some social and conceptual conflict starts to emerge as
students debate whether the energy of the system can be zero. First, the various
“starts” by Bob, Dan, and Al (lines 4, 6, 7, 8, 9, 13, 16) correspond to different
potential entry points, none of which take hold unchallenged. Indeed, the starts of
many of the utterances (lines 6, 9, 13, 16, 17, 18) serve to counter the previous
utterance either through direct challenge or through proposing a different path. It is
unclear in some instances if the disagreements are based on failures to actively listen
or on intentional disagreement. Either way, though, this initial volley of embodied
oppositional stances constitutes a tense exchange among the group members.
To support this conclusion, we now walk through the discourse line by line. In
lines 8 and 10, Dan suggests that looking to the harmonic oscillator might provide a
clue because in that case, the ground state starts at n=0. Al immediately follows with
an argument that even for the harmonic oscillator, the lowest energy state uses a value
of n=1, but is just referred to as the “n=0” state. The “right, but...” on which Al begins
signifies that what follows is likely to challenge Dan’s utterance (Goodwin &
Goodwin, 2002). In response, Dan and Chad talk over each other to correct Al’s
34
reasoning. Even before their simultaneous turns of talk are over, Al, leans back,
gestures, and rapidly says, “okay. yeah yeah yeah yeah, you’re right, but…” Al’s
response to Chad and Dan, (lines 18-24), is punctuated with continuous gesturing
using both hands. During his utterance, Al taps his paper repeatedly with his pen and
pushes his paper towards the center of the group, thereby offering and loudly
animating an object (the paper) around which the group can converge their attention.
Al finishes his utterance using large gestures that involve almost fully extended arms.
Al’s expanding embodied counters to his group members’ challenges, his subsequent
pushing for his group member’s attention to his paper, and his final use of extended
gestures demonstrate a growing tension within this short interaction (Goodwin, 2007;
Goodwin & Goodwin, 2002). His utterance acknowledges that he made a mistake,
concedes that Dan and Chad are right about the mathematics, but follows that with
bringing the conceptual and mathematical substance back to the square well problem.
So, the tension building up here emerges through conceptual and social
conflict. Their search for an explanation is bound up in challenging one another and
saving face. We can imagine less charged interactions in which a group tries to
unpack how the square well relates to the harmonic oscillator, and why a substitution
of n=0 makes sense for one but not for the square potential, in a way that does not
continuously put the speaker’s face at risk. But this space, as currently constructed, is
one in which physics knowledge also serves as a tool for establishing superiority. In
the rapid exchanges that challenge previous ones, repairs, and acknowledgments of
35
who is right and who is wrong, we see coupled conceptual and social dynamics as
contributing to tension building up within the group.
Segment 3/6: Status negotiation at the expense of a peer
36
We see continued building of tension. The conflict in the group takes on
conceptual, epistemological, and social components. Bob and Chad begin to
collaboratively suggest that a particle must always have some sort of energy, and
hence zero energy would suggest the non-existence of the particle. Chad makes an
analogy to a “bushel of no apples.” Al responds to Chad’s statement with “NO::” with
the strength of his disagreement indicated by loudness and vowel elongation (line 45).
His gestures add to the explicitness of his disagreement: when he mentions “a ball in
a well,” (lines 47-48) he shapes his right hand into a loose fist which he raises up, and
then allows to fall loudly on the table. So, his talk and actions embody opposition to
37
Chad’s suggestion, with the forcefulness of the opposition bringing a social and
emotional component to what on paper looks like a conceptual disagreement about
the possibility of a zero-energy particle.
Al also introduces an epistemological disagreement to the discourse. He adds
to the authority of his counter to Chad when he takes a position of privilege as
someone who is able to interpret what the tutorial is asking for (in line 45, “What I
think they're saying is...”) (Sullivan & Wilson, 2015). He says the tutorial is asking
for “the difference,” which from the lines 45-48 and subsequent talk, we take to mean
the difference between a classical and a quantum particle, with respect to whether a
particle can have zero energy. Unlike Bob and Chad, for whom the answer is an
assertion about whether a particle can have zero energy (without specifying what kind
of particle), Al wants different answers for a classical vs. a quantum particle. And
while Bob in line 49 takes up and adds onto Al’s assertion about classical particles,
acknowledging that a classical particle “can just be sitting there” presumably with no
energy, he does not take up Al’s suggestion to separately consider classical vs.
quantum particles, as we’ll see in the next section. In any case, Al’s positioning of
himself as uniquely able to interpret the tutorial’s intent and his forceful conceptual
disagreement with Chad generates tension in the group, as evidenced by Bob
smirking at Ed during line 45.
Segment 4/6: Limited collaboration turns the conversational focus
In this segment, the tension continues. Al cuts off Bob with a reassertion of
what the tutorial writers are looking for, but then Bob and Chad do not respond to
38
Al’s bid. They instead pursue another line of reasoning, with no participation from
Al.
In line 53, Al continues explicating his interpretation of the tutorial question
doing so, he cuts off Bob in line 52. Bob’s response in line 56 addresses neither the
39
interruption nor the substance of Al’s bid. Instead, he starts engaging in a new
epistemic activity, trying to remember some information from an authoritative source
(“physics book”) about the minimum speeds that objects of different sizes can have.
Chad takes up Bob’s line of reasoning by requesting (line 60) and then affirming (line
63) Bob’s clarification of what “minimum speed” means. In summary, this segment
of discourse is non-collaborative between Al and Bob, both in a social sense (Al cuts
off Bob, Bob ignores Al’s ideas and Al is shut out of conversation) and in an
epistemological sense (Al and Bob are engaged in two different, non-interacting
epistemic activities). This non-collaborativeness, we claim, sustains the earlier
tension.
Segment 5/6: Epistemological statements close mathematical topic
In this segment, the epistemological conflict is at least temporarily resolved as
the entire group, including Al, takes up Bob’s bid at the end of segment 4 to explore
what linear algebra, which is the mathematical formalism used in quantum
mechanics, has to say about the possibility of a zero energy particle. However, the
tension continues until, at the end of the segment, Al makes a bid for taking an escape
hatch.
40
41
Starting immediately in line 69, the group takes up Bob’s suggestion to
discuss whether an eigenvector/eigenvalue can be zero, giving tacit approval to this
direction. Al first responds, saying that “it's zero isn't an eigenvector.” His tone lacks
inflection and his words are well-enounced, indicating some confidence in his
response. In line 74 Chad interrupts Al to finish his statement. Al initially responds
with “eigenvector” but Chad answers with “eigenvalue.” Chad promptly opposes Al’s
idea, cutting him off before he can voice it. Al responds in kind, taking the floor from
Chad to apparently summarize Chad’s point for him, “'cus then any vector could be
an eigenvector.” We see Al as espousing Chad’s position for him, rather than Al’s
own potentially changing position, because Chad is arguing for eigenvalues not being
able to be zero (line 74) while Al is in favor of eigenvectors being unable to take on
zero value (line 69). Accordingly, Bob asks for clarification from the group after Al
finishes Chad’s utterance with, “oh, so it can't be an eigenvalue?” Bob’s request for
clarification is met with forceful opposition from Al. During his utterance in line 77,
42
Al shakes his head, drops his pen and hits the table with his pointer finger repeatedly
while reiterating his point (finger pointing into the table). The drawn-out “no” by Al
and his embodied response as a whole, highlights his opposition to Bob’s suggestion
that it is an eigenvalue that cannot be zero. Al’s initial statement in line 69, and his
reframing of this statement in line 77, indicate that his belief that zero cannot be an
eigenvector is somewhat stable through this piece. This further suggests that what Al
is doing in line 75 when he finishes Chad’s utterance for him, is taking away
opportunities to participate from Chad. Al voices an opinion for Chad that contradicts
his own views, only to strongly push back against this opinion in his next turn.
After Al suggests that zero can be an eigenvalue in line 77, both Chad and
Dan disagree with him. Opposition in the group is apparent in these lines, as three
subsequent speaker turns begin with “no,” (Goodwin & Goodwin, 2002). The group’s
opposition might be enough to make Al reconsider his point of view, and he reiterates
the rest of the group’s position with “then zero can be an eigenvector.” It now almost
appears as if Chad, Dan and Al all have settled on eigenvalues being unable to be
zero. Bob then takes the next steps in the group’s reasoning, attempting to make
inferences based on the group’s apparent position, “so then if zero can't be an
eigenvalue and if the way you--” However, Bob is cut off by Al who responds with
“it's whatever one that makes it like trivial.” This utterance allows Al to superficially
acknowledge what condition the group’s solution must satisfy, without actually
identifying which possibility—no zero eigenvectors or no zero eigenvalues—satisfies
43
that condition. Al is also indicating, or making a bid, that the issue has not been
resolved within the group.
In line 92, Bob tries to synthesize the group’s position. However, Al’s
response to Bob is not one that incites further discussion but makes a bid to close
down the conversational topic altogether—a bid for an escape hatch, as we’ll argue
below. Al proposes a close with “so we can say linear algebra,” to which Dan agrees
with “because math”—an adjacency pair type (Schegloff & Sacks, 1973). Al’s first
pair part is a joke about the mathematical conclusion reached by the group, to which
Dan responds in kind, with the joke, “because math.”
The rest of the group implicitly agrees to close through their laughter and their
openness to Al’s redirection of conversational topic, discussed in the next section
below. The “because math” joke reiterates the group’s epistemic stance that
mathematics was the preferred place to look for warrants for their arguments while
also acknowledging, through humor, that their mathematical “resolution” is perhaps
not fully satisfying. For the argument of this paper, however, the key point here is that
the joke relieves tension in the group. The group members smile, laugh and lean into
the table.
This episode is not an escape hatch simply because the group interactionally
achieved relief of tension. It’s an escape hatch because (1) the conflict that emerged
was multifaceted (conceptual, social, emotional, and epistemological) in nature, (2)
closing move(s) contribute to relief of the tension as an interactional achievement,
but (3) without resolving the conceptual issue and/or epistemological issues that
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helped produce the tension in the first place—in this case, reconciling between
competing ways of understanding why a quantum particle in a box cannot have zero
energy.
Segment 6/6: Coming to an uncertain conclusion
With the escape hatch having been taken, the discourse gets a fresh start. Al
restates an approach he had introduced back in segment 1, but this time the group
takes up the approach in a collaborative way.
45
46
Al’s emphasis on “qualitative” in line 98 suggests that he might be thinking of
his reasoning based on the uncertainty principle as distinct from the mathematical
reasoning they have been pursuing for the last few minutes. The group shows their
support for the need for a conceptual response by allowing Al to complete this
relatively long statement without interruption (lines 98-100), after which they begin
to collectively explore what the uncertainty principle may offer. Bob is the first to
offer support to Al’s suggestion with his exclamation (line 101). Al then concedes
47
that he doesn’t know how to explain his solution further (line 105). Bob and Chad
subsequently continue the line of reasoning, negotiating what would happen to the
particle’s energy, momentum and position. Although Bob and Chad make repairs to
each other’s contributions, they still build on each other’s reasoning (and on Al’s and
Dan’s)—a collaborative discussion that contrasts with the lack of collaboration in
segment 4. Although the students don’t reach a full resolution using qualitative
reasoning centered on the uncertainty principle, they make progress and end up
connecting that qualitative reasoning with the mathematical ideas they had been
discussing earlier. So, as we discuss later in more detail, the escape hatch in this case
provided tension relief that enabled the group to restart their discussion in a way that
helped them make progress addressing the question at hand.
Episode 2: “Can we define”
Our second episode comes from later in the same session. Here, the “escape hatch” is
a locally closing segment of conversation in which the students re-interpret the
worksheet question in a way that allows the group to move on to the next question
without resolving the preceding conceptual disagreement.
The episode begins with Chad reading the question out loud. Al then proposes
considering points on the wave (transcript lines are relabeled, starting from 1.)
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Segment 1/4: Which speed and which frame?
Al (2-4) begins talking about the physical situation, noting that the motion of
individual points on the wave could be interpreted in terms of speed. Bob follows up
with a clarifying question (line 5). Bob's question seems to be a bid to first make
meta-level sense of the task, i.e., what is the tutorial asking us to do? The ensuing
discussion raises two possibilities for “defining a speed for the wave”: defining the
horizontal speed with which the wave propagates (Dan, line 6; Chad, line 7), or the
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vertical speed with which particles/point(s) on the wave move up and down (Al, lines
2-4).
Segment 2/4: Oppositional Stances to Defining “Speed”
50
Chad and Bob (lines 18-25) co-construct a conceptual approach to the
problem of decomposing the standing wave into its rightward and leftward traveling-
wave components, and taking the speed of one of the components. Al contests their
claim, using his thumbs to demonstrate a similar situation, one in which their
decomposition idea does not make sense. Al may be suggesting that, because using
decomposition to define speed applies only to the particular case of a standing wave,
it might not be a valid way to define wave speed more generally. Bob, Dan, and
Chad emerge as aligned with one another, and dis-aligned with Al, in repeatedly
making the same point to Al that the case under consideration is a standing wave. Bob
and Dan add additional epistemological layers to their disagreement through their
choice of words of, “the whole point is it’s a standing wave,” and “they’re saying it’s
a standing wave,” which invoke external authority.
So here, the initial conceptual disagreement on defining speed gets coupled
with conceptual, epistemological and social components, in that one group member’s
questioning of how generalizable a line of reasoning should be is not interactionally
taken up by the other group members. Instead, the other group members provide
repeated, distributed, and epistemologically weighted protests against one member’s
bid for an epistemological reconsideration of what counts as a valid definition.
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Segment 3/4: Al’s questioning is shut down
52
In this excerpt, the epistemological+social conflict grows, in Al reiterating his
concerns over the rest of the group’s reasoning and the other members continuing
53
their distributed, embodied counters to Al’s concerns in a way that shuts Al out of the
conversation after line 59.
Al begins the excerpt by expanding on his bid to have the group consider the
epistemic validity of their reasoning. During Al’s utterance, both Chad and Bob smirk
at Al and each other, both exchanging glances with Dan. Bob and Chad talk over each
other to respond to Al’s comment, elaborating on why the components of a standing
wave have the same speed but not addressing Al’s point that in other physical
situation the components could have different speeds. Chad responds that “speed
doesn’t have a direction,” which while technically correct directly contradicts his
gesture animating the wave components in Segment 2, lines 20-23. Al conveys a
sense of frustration when he responds with, “No:. No, no, no,” and goes on to
reiterate, “what I’m saying,” a phrase Al has used in back-to-back turns. His repeated
use of this phrase indicates that he may feel his point is not getting across, but he
concedes before finishing his explanation. Pronoun use in his utterance, clearly
marking his utterance as his own (“what I am saying”) and marking ideas he is
challenging as belonging to the group (through the use of “you”), also indicate his
sense of separation between himself and the group.
Chad’s utterance (lines 60-64), immediately following Al’s initial protest and
then resignation, seems directed at Al: Chad glances away from Al only once during
lines 60-64, and fully removes his gaze from Al only when Bob begins his utterance
in line 66. However, Al is unengaged during this time, looking and leaning into his
paper. His physical positioning and actions differ starkly from the rest of the group,
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who are upright and attending to Chad. Bob and Chad go on to co-construct a story
about the speed and reflection of the components of a standing wave (lines 60-78). In
lines 69-75, Chad provides an embodied complement to Bob’s reasoning, gesturing
with Bob’s utterance. The high level of collaboration between Bob and Chad here
makes their misalignment with Al even starker.
Thus in this segment, we see a further construction of alignment between Bob,
Chad, and Dan, and their growing dis-alignment with Al, with interactional markers
that the alignment/dis-alignment with respect to conceptual and epistemological
approach to the question also has affective and social dimensions.
Segment 4/4: Escaping continued tension: undefinable speed
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In this segment, the coupled conceptual, epistemological, and social
divergence between Al and the rest of the group continues (contributing to tension)
until playing on the wording of the tutorial provides a means of closing the
discussion—the escape hatch.
This segment functions as a conversational closing, initiated with Ed’s bid in
line 79. His utterance starts with the hedge word “so”, indicating that he might be
wanting a shift from the earlier discussion (Bolden, 2009); and in contrast to the
earlier discussion, Ed refers back to the wording of the tutorial question, “can you
define the speed?” Through reference to what opened the conversation, Ed
demarcates this conversational point as an opportunity to close (Schegloff and Sacks,
1973).
Ed’s bid for the group to find an answer is met with the group revealing
uncertainty in their tentative conceptual solutions and a continuance of the tension
between Al and the rest of the group. Chad’s opening and uncertain “uhh” and his
upward inflection at the end of his utterance suggest that Chad is unsure about
whether “phase velocity in one direction” is a correct resolution. Bob displays similar
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uncertain feelings of the group’s conceptual footing as he smiles, shakes his head and
quietly says “I don’t know.” Al then restates his earlier stance towards Chad’s
conceptual solution. In response, Bob and Chad collectively position Al as now
responsible for finding the solution (lines 84-86).
As Al completes his utterance in lines 87 and 89, gesturing towards his paper
while noting that he doesn’t think there is one, Chad looks to his own paper and
exclaims, “oh! It’s CAN we define!” This utterance marks a stark shift in the group’s
affect; the earlier tension gives way to mirth. Al voices his affirmation and the rest of
the group laughs and leans into the table. Chad laughs as he leans over his paper, “no.
no, we cannot.” Similarly, Bob voices a drawn-out, lilted “no:” sharing in the group’s
humor. Throughout the discussion, the students were implicitly framing their task as
finding an appropriate speed of the wave, proposing and opposing many different
definitions along the way. However, Chad’s comment (line 89) allows the group to
re-frame the discussion as addressing whether it is possible to define a speed, a
reframing supported by the “Can we define” wording of the tutorial.
For this conversational closing to be an escape hatch, an underlying
conceptual disagreement must remain unresolved. Al may have reached some sort of
conceptual resolution, his principled argument against the decomposing the standing
wave into components in order to define speed. For Bob and Chad, by contrast, it’s
not evident that a conceptual resolution has been reached. Chad keeps suggesting the
decomposition (phase velocity) line of reasoning until the re-framing at the very end.
Additionally, Chad seems to be responding more towards the wording of the tutorial,
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rather than some notion that there is not a well-defined speed, as Al might be
suggesting. Re-framing the "can we define" wording acted as a pivot for the group,
helping the conversation to close and to relieve the tension created earlier.
Episode 3: Agreeing to disagree
The students in the group are Al and Ed (from episodes 1-2 above) now joined
by Karen and Larry. When we held the focus group meeting, Al, Ed, and Larry were
enrolled in the second semester of a quantum mechanics course for physics majors.
Karen had taken these courses the year before. In this Episode, the students are
addressing the question:
Consider the hydrogen atom. One student described the
relationship between energy and position by saying ‘The hydrogen
atom is in a higher energy state when the electron is farther from
the nucleus. Do you agree or disagree? Why?
In what follows, the students’ are unable to reconcile their perspectives on
whether higher energy means farther from the nucleus.
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Segment 1/4: Two conceptual ideas posited against each other
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Conversational patterns emerge here that are largely sustained. These include
cutting each other off and talking over one another (lines 6-7, 9-10, 6-17, 23-24, 32-
33), with Larry and Al the primary interlocutors. Al’s and Larry’s gazes tend to stay
on each other, and not fall on Ed or Karen, even when Karen explicitly adds on (lines
10-11) to Larry’s idea. This further pushes Ed and Karen to the periphery of the
conversation.
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Considering the content of Al and Larry’s discussion, we see each speaker as
attempting to reason out the question as individuals rather than building on each
other’s’ ideas: the individual speakers engage in self-repairs (lines 5-6, 16, 18-20, 21-
23, 25-26, 29-32, 33-36), but rarely engage each other's’ lines of reasoning, e.g.,
Larry responds to Al’s question about angular momentum by talking about his own
thinking about distance, (Barron, 2003). Hence, through the (mild) social conflict of
weakly collaborative discourse, conceptual conflict also emerges, with Larry and Al’s
conceptual approaches (orbitals versus electron shells) posited against each other.
Segment 2/4: Sense-making with limited participation
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The interactional patterns seen in the last section continue. We see
differentiation of the social conflict associated with these patterns, in that Karen and
Ed are interactionally positioned with fewer opportunities to contribute, while the
main contributors (Al and Larry) continue to interact weakly. In order to make a
better-warranted case that access to participation is inequitable, we tabulate each
student’s contributions to the discussion (in Segments 1 and 2 of Episode 3) in the
table below.
Types of Contributions Karen (lines) Al (lines) Larry (lines)
Short affirmations 2 (13, 17) 2 (24, 28) 0
Cut-off by others 3 (10-11, 37, 48) 1 (5-6) 4 (7-9, 15, 29-32, 41-42)
Extended 1 (10-11)
6 (5-6, 16, 18-20, 33-36,
38-20, 43-44)
8 (7-9, 15, 21-23, 25-27,
29-32, 41-42, 45-47, 49-50)
Total number of turns of
speech 5 (10-11, 13, 17, 37, 48)
8 (5-6, 16, 18-20, 24, 28,
33-36, 38-20, 43-44)
8 (7-9, 15, 21-23, 25-27,
29-32, 41-42, 45-47, 49-50)
TABLE 2.1: Breakdown of types of contributions made to the discussion by Karen, Al, and Larry.
Number of contributions, per type, is shown in bold, followed by the associated line numbers in the
transcript. “Short affirmations” were contributions of the form, “yeah” and “sure,” or were brief
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repeats of another person’s utterance, such as Karen’s “not necessarily,” (line 15). “Cut-off by others”
indicate lines where the speaker was interrupted by another. ‘Extended’ utterances are turns in which
the speaker was able to connect at least two ideas together, or finish a complete thought.
In table 2.1, we can see that Karen had fewer opportunities to make extended
utterances, (1 in comparison to 6 for Al and 8 for Larry). Half of Larry’s utterances
were cut-off, and almost all of Karen’s. Even with the variance associated with the
number of contributions, what stands out is that Al and Larry are simply allowed to
contribute more before being cut-off, if at all.
The positioning of Ed as peripheral member also becomes more apparent. In
Segment 1, Ed contributed several times, but seemed to fade out as the conversational
dominance of Al and Larry became more cemented. The current excerpt begins with
no contributions from Ed. Towards the end, however, Larry’s interruption of Karen in
Line 49-50, functions as taking away Karen’s opportunity to participate while giving
the floor to Ed.
Additionally, we see continued evidence of the social and conceptual conflict
associated with the two weakly interacting conceptual lines of reasoning (Rochelle &
Teasley, 1995). Al spends three utterances, (lines 33-36, 38-40, 43-44) explicating his
position, with Larry only acknowledging those contributions with “yeah” and minor,
yet non-engaging, disagreements (line 41-42). Al seems to feel the interactional and
conceptual conflict, in that his tone becomes more forced through this segment and he
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feels the need to further demarcate his position through the preface of “what I’m
saying is,” (lines 33, 43).
Segment 3/4: Status negotiations among members and their contributions
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65
66
Within the continuing interactional patterns of weak interaction (until the
conversation settles into a discussion of the Fermi sphere, lines 79-100), status
negotiations among peers become more explicit and forceful.
Karen first introduces a new conversational topic, the Fermi sphere (line 53-
54). However, Larry’s response, instead of building on or continuing that thread,
physically and conceptually draws joint attention back to the graph he drew. His next
utterance serves to challenge Karen’s understanding with “you know what this is,
right,” positioning himself as more knowledgeable than Karen (Sullivan & Wilson,
2015). Through this interaction, Karen seems to experience growing discomfort,
which she expresses through nervous laughter, elongated “no’s” and blushing. It’s
evident that Al also experiences the growing tension between Karen and Larry, as he
closely follows the back and forth, smiling when Larry asks Karen, “you know what
this is, right?” Ed also attends to this interaction, looking back and forth between
Karen and Larry as they speak. Although we cannot infer what Ed is feeling in
response to the exchange, it is apparent that he is noticing it. In any case, Ed offers a
possible resolution to the question (line 63) after which we don’t see him making
further utterances here, even though the discussion continues for several more
minutes.
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After Larry’s fairly didactic utterance explaining “why we do quantum
mechanics” (line 76-78), Karen reignites her conceptual contribution of the Fermi
sphere. While explaining it, Karen notes twice that she didn’t realize the “guys
haven’t learned that yet,” explicitly positioning her background against those of the
other group members. She pushes her paper towards the center of the group, and
looks between the paper and Larry as she speaks, giving a sense that the explanation,
and possibly also the positioning moves, are really intended for Larry. Through these
embodied actions, she physically and relationally vies for more leverage, perhaps
unconsciously, to ‘buy’ her way into meaningful participation in the discussion.
However, Karen’s burst of participation is short lived, as Larry in turn positions her
idea as irrelevant (lines 96, 97), noting that it has nothing to do with their joint query
of “higher positions.” More importantly, he also physically positions her and her
ideas by turning his body and gaze away from her and to the rest of the group as he
finishes speaking, again putting her to the periphery of the conversation.
These status negotiations between Karen and Larry are not lost on the other
members, Al and Ed. Al smiles as Karen comments about what the others haven’t
learned. Ed moves from looking down and away to quickly looking between Karen
and Larry around the same time.
As before, we see tendrils of epistemological and conceptual conflict
emerging from largely social conflict and becoming more striking with time, with
Karen bidding for her knowledge to be considered authoritative but with the group
continuing to privilege certain members’ conceptual contributions over others. The
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emergence of multidimensional conflict brings a layer of tension, as the students
witness and react to the conflict unfolding in the group.
Segment 4/4: Agreeing to disagree
Ultimately, the tension associated with Al’s and Larry’s two weakly
interacting lines of reasoning, which is correlated with the epistemological conflict
associated with vying for epistemic authority (Greer, Jehn, & Mannix, 2008), appears
to win out over striving for conceptual resolution. Al begins the excerpt with a bid
for joint sense-making; he notes what “we should say,” and goes on to explain with
“because when you see.” However, Larry talks over him and cuts him off with his
bid to close through re-framing their discussion as one that doesn’t need group
consensus.
Lines 103-104 constitute the adjacency pairs that linguistically mark the close
of the conversation. In line 103, Larry asks the group if they will go along with the
close. Karen’s drawn out “ye:ah” shows some finality in her response. Larry’s last
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line acknowledges Karen’s agreement (Al has already agreed and Ed is no longer
involved), and aims to serve as the final turn in the conversation. So, as in the
Lampert et al. (1996), example, “agreeing to disagree” serves as an escape hatch;
changing the rules about what counts as an acceptable answer, with consensus no
longer required for acceptability, sidesteps resolving the conceptual disagreements,
renders moot the negotiations over epistemic authority, and relieves the tension.
Discussion and Instructional Implications
Entanglement of cognitive and social tensions
The literature review above showed that, while many studies focus on a
particular kind of conflict in student interactions (e.g., social, epistemological,
emotional, conceptual), few studies simultaneously focus on multiple kinds of
conflict and the relations between them. Building on work from organizational studies
showing correlations among different kinds of conflict and on the small body of work
in science and mathematics education showing the simultaneous presence of multiple
kinds of conflict in small-group work, our three episodes support the conjecture—
suggested by but not foregrounded in prior studies—that the different types of
conflict are not merely co-present but deeply entangled at a fine time scale. In
particular, group interaction can become tense as conceptual, epistemological, and
social conflicts emerge and build in group discourse.
This entanglement has a couple of implications for researchers. First, the ease
with which conceptual, epistemological, social, and emotional dynamics feed on each
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other invites close attention to learning environments and interactional patterns that
prevent such entanglement—specifically, that allow conceptual and epistemological
disagreements to emerge (and to be addressed) without overwhelming social conflict
or tension. Of course, experienced facilitators of small-group work already know to
be on the look-out for unproductive group dynamics and to try to set expectations for
productive argumentation (e.g., “accountable talk” moves (Michaels, O’Connor, &
Resnick, 2008) emphasized in some classrooms). We suggest that the ease with which
different types of conflicts can become entangled can help explain why structures
such as accountable talk moves don’t always “work,” and hence, why other pre-
emptive and in-the-moment interventions—as discussed below—might sometimes be
needed. Second, even if a researcher is ultimately interested in one type of conflict
(e.g., epistemological), the degree of entanglement we documented among the
different types of conflicts suggests that deeply understanding the genesis and nature
of one type of conflict will require at least some research attention to the other types.
For instance, in the “because math” episode, we simply can’t understand how the
epistemological conflicts arose, sustained, and then ebbed without also attending to
the social conflict and tension with which the epistemological conflict was coupled.
Escape hatches: a tension-relieving interactional achievement
Our episodes start to chart the landscape of escape hatches— resolutions of
tension interactionally achieved through closing a conversation without resolving the
conceptual conflict that helped to produce the tension in the first place. To be clear,
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we did not “discover” escape hatches: Lampert et al. (1996) focused on how Sam and
Connie decided to “agree to disagree” to escape their tense conversation. And
instructors probably notice similar phenomena. Our contribution is to recognize
“agreeing to disagree” as a special case of a more general “escape hatch”
phenomenon that is also achieved in other ways and to begin exploring the
commonalities and differences among different instantiations of escape hatches.
A key commonality we documented is that, perhaps partly because different
types of conflict are entangled, the escape hatch relieves all the types of conflict
instead of just the tension. We also documented how adjacency pairs, a conversation-
closing structure documented by sociolinguists across a broad range of interactions,
play the same “closing” role in an escape hatch.
The differences in instantiations of escape hatches we documented—agreeing
to disagree vs. epistemological humor about the explanatory power of math (“because
math”) vs. exploiting the wording of the tutorial question—likely only scratch the
surface of the multiple ways in which escape hatches could play out. For now, our
point is that it can do work for instructors to see these (and yet-to-be-documented)
disparate patterns of turn-taking (or not), gesturing, and adjacency pairs as different
instances of a thing, an escape hatch. We now take up this point.
Instructional implications. Although we’ll discuss specific instructional
moves below, we do not think these moves are our main take-away. Instead, first and
foremost, we believe that our work can help instructors develop an “escape-hatch
lens”—an attunement to noticing when, how and why students might be taking an
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escape hatch. This attunement, we suspect, consists partly of knowing about specific
analytical tools such as adjacency pairs (for noticing escape hatches) and markers of
escalating tension such as extended gestures, changes in paralinguistic features,
students “ganging up” on another student, discussions lacking mutuality, relative
positioning of students and their ideas, failures to come to shared understandings,
repetition of one’s own idea, and so on. But even more central to the escape hatch
lens, we suspect, is a holistic, intuitive sense of how conceptual, epistemological, and
social can emerge and give rise to tension. This tension and multidimensional conflict
may be relieved in a way that doesn’t address the conceptual issue at hand. As
researchers, once we became attuned to conflict escalation, tension, and escape
hatches, we started seeing them everywhere, including in prior literature such as
Lampert et al. (1996). We suspect the same will be true for instructors who regularly
facilitate small-group work.
An escape hatch lens includes not only noticing but also interpreting the
socio-cognitive phenomena leading up to and constituting an escape hatch. For
instance, consider conversation closer “because math” (episode 1). An instructor
listening in might think that “because math,” though humorous, represents a
consensus epistemological view that a mathematical answer to the question is
sufficient. Or, foregrounding the humor, an instructor might think that “because
math” is nothing more than joke. However, an escape hatch lens suggests an
interpretation in which the epistemological stance toward mathematics is both half-
serious and half-joking and serves to relieve tension to allow the group to move on.
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Similarly, when the group in episode 3 decides to answer the question “Can we
define,” an instructor might think the students are reading the tutorial too literally and
might therefore suggest that they instead address “How can we define…” An escape
hatch lens suggests, by contrast, that the literal reading was a tension-relieving move,
not the group’s original or default reading of the tutorial question, and that telling the
students to continue the discussion would likely lead to a renewal of the conflict that
gave rise to tension and motivated the escape hatch.
Indeed, from our own (mostly minimalist) facilitation of episode 3, we
observed what can happen when an instructor directs a group to return to the
conceptual conflict from which they escaped. The facilitator did notice the discussion
becoming quite tense and the escape hatch that the students took (agreeing to
disagree). However, the facilitator perceived the conceptual idea of potential energy
as a sticking point for the group, impeding their progress and resulting in their
inability to come to a consensus. As such, the facilitator intervened with “So I think
we're not, concerned as much with potential energy here, just if my electron has more
total energy, it's in a higher like n, does that mean it's further from the nucleus?”
However, the intervention was largely unsuccessful. The group obliged the implicit
instruction to continue talking, but they did so half-heartedly. Larry seemed sarcastic,
Karen provided an answer of “not necessarily” but provides no explanation upon
request. Ed simply said “not sure.” An escape hatch lens makes this result
unsurprising and suggests an instructional need to address the sources of escalating
tension, which is coupled with why the conceptual space took the twists and turns that
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it did. So while the intent of the facilitator was to turn the students back to their
‘conceptual’ discussion about potential energy, the move also turned the students
back to an emotional tense conversation about potential energy.
Productivity of conflict and escape hatches in group discourse. Although
we have spent a good deal of this paper arguing for the entanglement of different
types of conflict, we do not aim to suggest that any/all conflict is bad. Indeed, an
underlying premise of collaborative group-work often assumes that some sort of
cognitive conflict will emerge in argumentation, potentially leading to a deeper
understanding of the material being discussed. The point we are trying to make is that
different types of conflict can be deeply entangled, and so emerge together in group
discourse. So while a tutorial may encourage some conceptual conflict among
participants, it is also possible that tension as well as different types of conflict will
emerge with the conceptual conflict. We have provided some means of noticing
escalating tension, different types of conflict, and their entanglement. However, it is
up to instructors to decide if the tension and conflict they see in group-work is
productive, or worthy of an instructional intervention. Determinations of productivity
will likely involve considerations of local and global instructional goals. For example,
is it more important, in the moment, for the students in episode 3 to understand why
the Fermi sphere is irrelevant to the situation or is it more important to help the group
establish more equitable discussion norms? Both are certainly possible. It’s up to the
instructor to decide whether the different conflicts emerging need instructional
resolution.
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The same line of reasoning goes for when instructors are considering the
productivity of students taking escape hatches. We don’t want to suggest that
instructors should always push back against escape hatches. In episode 1, the
“because math” escape hatch relieved tension and opened space for the group to have
a collaborative, conceptually rich discussion addressing the tutorial question. By
contrast, in episodes 2 and 3, the escape hatches represented a more permanent escape
from resolving the conceptual issue.
Curricular design implications. The “Can we define” escape hatch
highlights that curricular designers should be aware that certain question wordings
can invite escape-hatch responses. We suspect that such invitations are ubiquitous.
For instance, instructors of large undergraduate lecture classes often pose multiple-
choice “clicker questions” for students to discuss in small groups before “voting”
individually on their answers. These clicker questions, sometimes adapted from
conceptual diagnostic instruments such as the Force Concept Inventory (FCI;
Hestenes, Wells, & Swackhamer, 1992), often include options such as “not sure,” or
“not enough information to tell.” The instructor has no way of knowing whether
students chose these answers for sound intellectual reasons or as an escape hatch from
a tense discussion.
Going meta on emotions
Influenced by our own instructional experiences and by studies of the role of
metacognition and empathetic listening in small-group problem solving (e.g.,
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Schoenfeld, 1992; Blumenfeld, Marx, Soloway, & Krajcik, 1996), we can quickly
rehearse “standard” instructional moves to respond to escalating tension and escape
hatches. These include having group members articulate each other’s ideas, verbally
or in writing; pausing the discussion for a few minutes to allow everyone to collect
and write down their thoughts; making space for non-participating group members to
share their ideas; and having students “go meta” on the source of their conceptual or
epistemological conflict. These types of interventions generally function to provide
some kind of awareness to students, although the awareness is generally limited to
content concerns and relative amounts of participation. As instructors, we routinely
help students become aware of conceptual ideas and epistemological approaches. But
this can leave the emotional aspects of interaction unaddressed.
And so we ask, why not help students become aware of their own emotions in
sense-making and group work? This is especially important given the entanglement
of emotional and cognitive dynamics in students’ sense-making as we showed above.
In fact, epistemological “shifts”—reinterpreting the tutorial question to redefine what
counts as an answer, agreeing to disagree, joking about the adequacy of a
mathematical response—seem to be recruited to manage the emotional content of
interactions, making emotion the driving factor and epistemology the tool to manage
emotion. With this in mind, we have sketched a general intervention that might work
well to head off or retroactively address escape hatches. The intervention is simple:
When a facilitator notices an escape hatch in action, or tense interactional patterns,
the instructor asks the students to take a moment to individually write how they are
77
feeling in that moment. Having done so, the students are then asked to think about
what conceptual ideas they are hearing and how they are deciding between them. This
reflective intervention prioritizes what students are feeling emotionally and provides
space for the students to connect their emotional states to the conceptual content. This
intervention may also provide a natural “re-set,” potentially disrupting problematic
conversational dynamics in the quiet-time. However, if those patterns re-emerge after
the intervention, an instructor might see that as an indication that a more explicit
intervention is needed.
Interactional dynamics can also develop more stable patterns over of the
course of a semester. This may mean that students settle into particular roles within
their group and that certain interactional histories develop between the instructor and
student(s). Judgements about how to intervene in group dynamics may therefore
change significantly over the course of the semester as the instructor and students
learn more about each other.
Conclusion
So far in this article, we have implicitly adopted the standard reason for
attending to emotions and social dynamics in group work: the type and degree of
conflict in group work can encourage or hinder cognitive development and shared
understanding (Van den Bossche, Gijselaers, Segers, Woltjer, & Kirschner, 2011;
Buchs, Butera, Mugny, & Darnon, 2004). As our parting shot, we advocate a more
radical take on the importance of escape hatches and related phenomena.
78
In students’ lives, managing tension and conflict in group problem-solving—
at home, at school, at work—is likely to be more important than their understanding
particular science concepts such as the relation between energy levels and proton-to-
electron distances in the hydrogen atom. We therefore propose that helping students
learn to notice tension, conflict, and other group-work-related emotions and
interactional patterns, go meta on them, and ultimately manage them, be a primary
goal of instruction, not just something we do in the service of helping students engage
in productive sense-making. In other words, we advocate re-conceptualizing the
classroom as a space where students learn how to grapple with disagreements and the
build-up of emotion in the face of those disagreements. With this work, we hope to
help support classrooms where students can deal with these emotionally-charged
disagreements without disengaging or truncating the discussion before reaching a
point of more clarity. For instructors to accomplish this broader goal, they need to
frame the classroom expansively, so that the conversational and emotional skills can
span beyond physics classrooms and include everyday and socio-political
conversations with friends and family. Now, more than ever, there is the need for our
classrooms to help students acquire the tools to engage with conceptually,
epistemologically, socially, and emotionally difficult issues in the world.
79
References
Alexopoulou, E., & Driver, R. (1996). Small-group discussion in physics: Peer
interaction modes in pairs and fours. Journal of Research in Science Teaching,
33(10), 1099-1114.
Aikenhead, G. S. (1985), Collective decision making in the social context of
science. Sci. Ed., 69: 453–475.
Amason, A. C., & Sapienza, H. J. (1997). The effects of top management team size
and interaction norms on cognitive and affective conflict. Journal of
management, 23(4), 495-516.
Amason, A. C. (1996). Distinguishing the effects of functional and dysfunctional
conflict on strategic decision making: Resolving a paradox for top management
teams. Academy of management journal, 39(1), 123-148.
Barron, B. (2000). Achieving coordination in collaborative problem-solving
groups. The Journal of the Learning Sciences, 9(4), 403-436.
Barron, B. (2003). When smart groups fail. The journal of the learning sciences,
12(3), 307-359.
Berland, L. K. and Reiser, B. J. (2011), Classroom communities' adaptations of the
practice of scientific argumentation. Sci. Ed., 95: 191–216.
Blumenfeld, P. C., Marx, R. W., Soloway, E., & Krajcik, J. (1996). Learning with
peers: From small group cooperation to collaborative communities. Educational
researcher, 25(8), 37-39.
80
Bolden, G. B. (2009). Implementing incipient actions: The discourse marker ‘so’
in English conversation. Journal of Pragmatics, 41(5), 974-998.
Bricker, L. A., & Bell, P. (2008). Conceptualizations of argumentation from
science studies and the learning sciences and their implications for the practices
of science education. Science Education, 92(3), 473–498.
Brookes, D. T., & Etkina, E. (2007). Using conceptual metaphor and functional
grammar to explore how language used in physics affects student learning.
Physical Review Special Topics-Physics Education Research, 3(1), 010105.
Brown, A. L., & Palincsar, A. S. (1989). Guided, cooperative learning and
individual knowledge acquisition. Knowing, learning, and instruction: Essays in
honor of Robert Glaser, 393-451.
Buchs, C., Butera, F., Mugny, G., & Darnon, C. (2004). Conflict elaboration and
cognitive outcomes. Theory into Practice, 43(1), 23-30.
Collins, A., Brown, J.S., & Newman, S.E. (1989). Cognitive apprenticeship:
Teaching the crafts of reading, writing, and mathematics. In L. Resnick (Ed.),
Knowledge, learning and instruction: Essays in honor of Robert Glaser (453-
494). Hillsdale, NJ: Erlbaum.
Conlin, L. D. (2012). Building shared understandings in introductory physics
tutorials through risk, repair, conflict & comedy.
Csikszentmihalyi, M., & Sawyer, K. (2014). Shifting the focus from individual to
organizational creativity. In The Systems Model of Creativity (pp. 67-71).
Springer Netherlands.
81
De Dreu, C. K., & Weingart, L. R. (2003). Task versus relationship conflict, team
performance, and team member satisfaction: a meta-analysis. Journal of applied
Psychology, 88(4), 741.
De Dreu, C. K., & West, M. A. (2001). Minority dissent and team innovation: the
importance of participation in decision making. Journal of applied Psychology,
86(6), 1191.
Derry, S. J., Pea, R. D., Barron, B., Engle, R. A., Erickson, F., Goldman, R. &
Sherin, B. L. (2010). Conducting video research in the learning sciences:
Guidance on selection, analysis, technology, and ethics. The Journal of the
Learning Sciences, 19(1), 3-53.
diSessa, A. A. (1993). Toward an epistemology of physics. Cognition and
instruction, 10(2-3), 105-225.
diSessa, A. A., Levin, M., & Brown, N. (edited volume, to appear in 2015).
Knowledge and interaction: A synthetic agenda for the learning sciences. New
York, NY: Routledge.
Dunbar, K. (1997). How scientists think: On-line creativity and conceptual change
in science. Creative thought: An investigation of conceptual structures and
processes, 4.
Erickson, F. (2006). Definition and analysis of data from videotape: Some research
procedures and their rationales. Handbook of complementary methods in
education research, 3, 177-192.
82
Savinainen, A., & Scott, P. (2002). The Force Concept Inventory: a tool for
monitoring student learning. Physics Education, 37(1), 45.
Flavell, J. H. (1979). Metacognition and cognitive monitoring: A new area of
cognitive–developmental inquiry. American psychologist, 34(10), 906.
Thornton, R. K., & Sokoloff, D. R. (1998). Assessing student learning of Newton’s
laws: The force and motion conceptual evaluation and the evaluation of active
learning laboratory and lecture curricula. American Journal of Physics, 66(4),
338-352.
Goffman, E. (1955). On face-work: An analysis of ritual elements in social
interaction. Psychiatry, 18(3), 213-231.
Goodwin, C. (2007). Participation, stance and affect in the organization of
activities. Discourse & Society, 18(1), 53-73.
Goodwin, M. H., Goodwin, C., & Yaeger-Dror, M. (2002). Multi-modality in
girls’ game disputes. Journal of pragmatics, 34(10), 1621-1649.
Greer, L. L., Jehn, K. A., & Mannix, E. A. (2008). Conflict transformation a
longitudinal investigation of the relationships between different types of
intragroup conflict and the moderating role of conflict resolution. Small Group
Research, 39(3), 278-302.
Hammer, D. (2000). Student resources for learning introductory physics. American
Journal of Physics, 68(S1), S52-S59.
83
Heller, P., & Hollabaugh, M. (1992). Teaching problem solving through
cooperative grouping. Part 2: Designing problems and structuring groups.
American Journal of Physics, 60(7), 637-644.
Heller, P., Keith, R., & Anderson, S. (1992). Teaching problem solving through
cooperative grouping (Part 1): Group versus individual problem solving.
American Journal of Physics, 60(7), 627-636.
Jefferson, G. (2004). A note on laughter in ‘male–female’ interaction. Discourse
Studies, 6(1), 117-133.
Jehn, K. A., & Mannix, E. A. (2001). The dynamic nature of conflict: A
longitudinal study of intragroup conflict and group performance. Academy of
management journal, 44(2), 238-251.
Jehn, K. A. (1997). A qualitative analysis of conflict types and dimensions in
organizational groups. Administrative science quarterly, 530-557.
Jehn, K. A. (1995). A multimethod examination of the benefits and detriments of
intragroup conflict. Administrative science quarterly, 256-282.
John-Steiner, V. (2000). Creative collaboration. Oxford University Press.
Johnson, D. W., & Johnson, R. T. (1979). Conflict in the classroom: Controversy
and learning. Review of Educational Research, 49(1), 51-69.
Jordan, B., & Henderson, A. (1995). Interaction analysis: Foundations and
practice. The journal of the learning sciences, 4(1), 39-103.
Hestenes, D., Wells, M., & Swackhamer, G. (1992). Force concept inventory. The
physics teacher, 30(3), 141-158.
84
Kelly, G. J., Druker, S., & Chen, C. (1998). Students’ reasoning about electricity:
Combining performance assessments with argumentation analysis. International
Journal of Science Education, 20(7), 849-871.
Kelly, G. J., McDonald, S., & Wickman, P. O. (2012). Science learning and
epistemology. In Second international handbook of science education (pp. 281-
291). Springer Netherlands.
Kutnick, P. J. (1990). A social critique of cognitively based science curricula.
Science Education, 74(1), 87–94.
Lampert, M., Rittenhouse, P., & Crumbaugh, C. (1996). Agreeing to disagree:
Developing sociable mathematical discourse. The handbook of education and
human development: New models of learning, teaching, and schooling, 731,
764.
Lave, J., & Wenger, E. (1998). Communities of practice: Learning, meaning, and
identity.
Lawson, C. A. (1956). An analysis of the process by which a group selects or
rejects ideas or beliefs. Science Education, 40(4), 245–253.
Lemke, J. L. (1990). Talking science: Language, learning, and values. Ablex
Publishing Corporation, 355 Chestnut Street, Norwood, NJ 07648 (hardback:
ISBN-0-89391-565-3; paperback: ISBN-0-89391-566-1).
Lovelace, K., Shapiro, D. L., & Weingart, L. R. (2001). Maximizing cross-
functional new product teams' innovativeness and constraint adherence: A
85
conflict communications perspective. Academy of management journal, 44(4),
779-793.
Lytra, V. (2009). Constructing academic hierarchies: Teasing and identity work
among peers at school. Pragmatics, 19(3), 449-66.
Michaels, S., O’Connor, C., & Resnick, L. B. (2008). Deliberative discourse
idealized and realized: Accountable talk in the classroom and in civic life.
Studies in philosophy and education, 27(4), 283-297.
Mortimer, E. F., & Machado, A. H. (2000). Anomalies and conflicts in classroom
discourse. Science Education, 84(4), 429–444.
Parnafes, O., & Disessa, A. A. (2013). Microgenetic learning analysis: A
methodology for studying knowledge in transition. Human Development, 56(1),
5-37.
Rosebery, A. S., & Puttick, G. M. (1998). Teacher professional development as
situated sense‐making: A case study in science education. Science Education,
82(6), 649-677.
Sacks, H., Schegloff, E. A., & Jefferson, G. (1974). A simplest systematics for the
organization of turn-taking for conversation. language, 696-735.
Schegloff, E. A., & Sacks, H. (1973). Opening up closings. Semiotica, 8(4), 289-
327.
Scherr, R. E., & Hammer, D. (2009). Student behavior and epistemological
framing: Examples from collaborative active-learning activities in physics.
Cognition and Instruction, 27(2), 147-174.
86
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving,
metacognition, and sense making in mathematics. Handbook of research on
mathematics teaching and learning, 334-370.
Shah, P. P., & Jehn, K. A. (1993). Do friends perform better than acquaintances?
The interaction of friendship, conflict, and task. Group decision and negotiation,
2(2), 149-165.
Shirouzu, H., Miyake, N., & Masukawa, H. (2002). Cognitively active
externalization for situated reflection. Cognitive science, 26(4), 469-501.
Simons, T. L., & Peterson, R. S. (2000). Task conflict and relationship conflict in
top management teams: the pivotal role of intragroup trust. Journal of applied
psychology, 85(1), 102.
Star, S. L., & Griesemer, J. R. (1989). Institutional ecology, translations' and
boundary objects: Amateurs and professionals in Berkeley's Museum of
Vertebrate Zoology, 1907-39. Social studies of science, 19(3), 387-420.
Stivers, T., & Sidnell, J. (2005). Introduction: multimodal interaction. Semiotica,
2005(156), 1-20.
Sullivan, F. R., & Wilson, N. C. (2015). Playful talk: Negotiating opportunities to
learn in collaborative groups. Journal of the Learning Sciences, 24(1), 5-52.
Roschelle, J., & Teasley, S. D. (1995). The construction of shared knowledge in
collaborative problem solving. In Computer supported collaborative learning
(pp. 69-97). Springer Berlin Heidelberg.
87
Van den Bossche, P., Gijselaers, W., Segers, M., Woltjer, G., & Kirschner, P.
(2011). Team learning: building shared mental models. Instructional Science,
39(3), 283-301.
Warren, B., Ogonowski, M., & Pothier, S. (2005). “Everyday” and “scientific”:
Re-thinking dichotomies in modes of thinking in science learning. Everyday
Matters in Science and Mathematics: Studies of Complex Classroom Events,
119–148.
88
Chapter 3: Learning with and about toy models in QM
Introduction
In this chapter, I study student sense-making with toy models in
undergraduate quantum mechanics. Toy models are highly simplified systems that
elucidate some physical mechanism underlying a phenomenon. For example, the
finite potential well model helps demonstrate tunneling of a quantum particle into
classically-forbidden regions. Toy models are also important in developing models of
potentially more complicated systems; aspects of formalism associated with a toy
model may also be important building blocks or serve as validity checks. For
example, raising and lowering operators in the context of the quantum harmonic
oscillator can be used to develop a coherent set of energy eigenstates in a model of
cavity-coupling to a two-level system (Nanda, Kruis, Fissan, Behera, 2004). For
physicists, toy models are not just something to know about to understand
fundamental quantum behavior, they are also tools for reasoning about and modeling
new situations. I argue that instructionally, toy models are already well positioned as
a collection of important models to learn about. However, they should also be treated
as tools for learning with (Greeno & Hall, 1997).
An important aspect of toy models in QM is that they are often associated
with particular iconic representations. By iconic representations I mean pictures or
graphs that are closely associated with the toy models. These pictures/graphs are
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iconic in the sense that they frequently show up in experts’ reasoning with and about
the toy models. They are ubiquitous in textbooks, curricula, and other classroom
artifacts.
In this chapter, I focus on students’ use of iconic graphical representations
associated with the infinite and finite well potentials and energy eigenstate
wavefunctions of quantum entities trapped in these wells. The focus on student use of
analogy through these iconic representation comes from an instructional interest on
being able to sketch wavefunctions. Wavefunctions are important for describing all
observable behavior of a system. Being able to qualitatively reason about the
behavior of quantum entities in a given system seems like an important first step in
developing an intuition about the system at hand.
Quantum physics has been one of the topics of physics explored within
physics education research. This collection of work and resources suggests that
fundamentals of toy QM models should be mastered before giving students
opportunities to use the models to understand novel situations. This work includes
curricular development (Singh, 2008; Zhu & Singh, 2012, Tutorials in Physics:
Quantum Mechanics3, QuILTs4, Intuitive Quantum Physics5), research on student
thinking (Bailey & Finkelstein, 2009; Bailey & Finkelstein, 2010; Emigh, Passante,
& Shaffer, 2015; Gire & Price, 2015; Johnston, Crawford, Fletcher, 1998; Kalkanis,
Hadzidaki, Stavrou, 2003; Ke, Monk, Duschl, 2005; Lin & Singh, 2009; Mannila,
3 https://depts.washington.edu/uwpeg/tutorials-QM 4 https://www.physport.org/examples/quilts/index.cfm 5 http://umaine.edu/per/projects/iqp/
90
Koponen, Niskanen, 2001; Marshman & Singh, 2015; McKagan & Wieman, 2006;
McKagan, Perkins, & Wieman, 2008a; McKagan, Perkins, & Wieman, 2008a;
Morgan, Wittman, & Thompson, 2003; Passante, Emigh, & Shaffer, 2015; Petri &
Niedderer, 1998; Singh, 2001; Singh & Marshman, 2015; Steinberg, Wittmann, Bao,
& Redish, 1999; Wittmann, Morgan, & Bao, 2005; Wittmann, Steinberg, & Redish,
2002), and conceptual surveys and inventories (Sadaghiani & Pollock, 2015;
McKagan, Perkins, & Wieman, 2010; Cataloglu &Robinett, 2002). This work has a
strong focus on patterns of student difficulties. This work also includes significant
research on student difficulties on concepts which seem essential to understanding
QM concepts (Sadaghiani, 2006; Bao & Redish, 2002). Some researchers have
further argued for the necessity of disrupting difficulties with classical mechanics
before moving onto quantum systems that build on classical concepts (Steinberg,
Wittmann, Bao, Redish, 1999).
I provide a complementary perspective. Through my analysis, I find that 1)
students are capable of applying these models before having developed a
sophisticated understanding of model, 2) the process of adapting the toy model
can lead to new understandings of the toy model and the systems represented
with the toy model. First modeling of these situations is often drawn directly from
these toy models (in various ways). It’s not always clear why, for students, they
intuitively draw on these models. Even though sometimes incorrect, immediate
adaptation of the toy model often directs student sense-making to areas or features in
one or both systems. Conceptual development then happens as students attend to
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additional features and regions in their representations. Sometimes this results in a
divergence of the student’s conceptual understanding of the two systems, i.e. students
develop an understanding as to why their initial, intuitive modeling of the situation
needed some refinement.
Background: Why create opportunities for adaptation of toy models and iconic
representations?
In this section, I argue that there is a need within quantum physics instruction for
creating opportunities for students to adapt toy models and iconic representations
towards understanding novel quantum physics scenarios. I structure this argument
through four sub-points:
1. Expert physicists adapt toy models to understand more complex quantum
physics scenarios
2. Adapting fundamental forms, concepts, and methods to understand novel
situations is a core aspect of disciplinary expertise
3. Current quantum physics instruction is inadequate in providing opportunities
for adapting toy models and iconic representations in QM to novel situations,
and
4. Current physics education research has underexplored student reasoning in the
context of adapting toy models and iconic representations in QM to novel
situations (which ties in with and reinforces #3 above).
I briefly explore each of these points below.
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Physicists adapt toy models to model more complicated quantum physics scenarios
In this section, I provide three examples of professional physics practice. In
each case, toy models are an essential means for developing new formalism. These
three reference cases include: the Jaynes-Cummings model, electron turnstiles, and
models of size-dependent band gaps in semiconductors.
The Jaynes-Cummings model describes coupling between a two-level atom
and a quantized light field. In more traditional, semi-classical treatments of cavity
coupling, the atom is treated as having quantized energy levels and the light is treated
classically. Developing the formalism of the Jaynes-Cummings model normally starts
with mapping out the coherent states of the system through reference to the quantum
harmonic oscillator (Daoud & Hussin, 2002). The harmonic oscillator provides an
ideal starting point of the formalism because the energy eigenstates are also
eigenstates of the lowering operator. The harmonic oscillator also ends up providing a
validity check in developing the Jaynes-Cummings model. For Daoud & Hussin,
there are points in their derivations that they take limits of a cavity coupling
parameter. Taking the limit is a means of showing they can recover aspects of the
harmonic oscillator formalism and therefore prove that their formalism agrees with
quantum cannon.
The goal of developing the Jaynes-Cummings model is to see any peculiar
quantum effects from treating the light field as quantized. In the semi-classical theory,
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Rabi oscillations eventually decay. However, the Jaynes-Cummings model shows
quantum revivals in these oscillations that arise from the discreteness of the photon
energy spectrum (developed through adaptation of the harmonic oscillator spectrum).
Another example comes from how modeling electron transport systems draw
on and adapt the finite quantum well toy-model. An electron turnstile is a means of
manipulating a single elementary charge. Ono, Zimmerman, Yamazaki, and
Takakashi developed a model of turnstile behavior in a single-electron transistor
(Ono, Zimmerman, Yamazaki, and Takakashi; 2003). To develop the model Ono et
al. piece together two finite wells as models for the source and drain gates. The finite
wells are then dynamically adapted; the “floor” voltage of the gates are modulated to
increase the probability of an electron hopping from one gate to the other. The current
can then be tuned through adjusting the modulation of the finite-well “depths.”
Band gap approximations by Nanda, Kruis, and Fissan provide a similar
example to the electron turnstile. This group also uses the finite well to develop a
model of a more real-world situation. Nanda et al. study the size-dependent band gap
of PbS and CuBr nanoparticles (Nanda, Kruis, and Fissan; 2004). To do so, they
model the electrons in the semiconductor as having an effective mass in a finite-depth
square well. The width of the potential comes from the size of the semiconductor and
its depth comes from the band gap energy(s) of the semiconductor. The band gap
energies that emerge from this model agree well with experimental measurements,
showing the importance of relying on the finite well toy model. This method stands in
contrast to those using effective mass approximations with no potential
94
approximation, or with an infinite well approximation. The finite well toy model has
more utility than the infinite well in describing the behavior of the materials of
interest to Nanda et al.
In summary, toy models are meant to be adapted to make sense of situations
that are arguably more complicated than the situations they represent. Complicated
can mean: modeling different types of entities (e.g. lattices), piecing together multiple
toy models, systems that involve interactions between entities, and dynamic models.
The ways in which toy models are used to make sense of these systems varies. In
some cases, the toy model is a way to make sure new developments reproduce
expected or empirical results. In other cases, aspects of the toy model become
explicitly involved in building new formalism and predicting new experimental
outcomes/results.
Problematizing toy model adaptation in QM instruction
As the examples above show, toy models are useful for helping develop
understanding in situations beyond those in which they were conceived. If instruction
in physics classes strives for authenticity, then students should be given ample
opportunities for adapting toy models. The question then becomes when is it
appropriate to position toy models as tools to learn with (adapt to new situations), not
just as representations to learn about.
In looking across common instructional materials for introductory QM
(Griffiths, 2016; Serway, Moses, & Moyer, 2004; Liboff, 2003; Singh, 2008; Zhu &
95
Singh, 2012, Tutorials in Physics: Quantum Mechanics6, QuILTs7, Intuitive Quantum
Physics8), I find the materials to be providing few opportunities for students to adapt
toy models and their representations to new situations. Most problem-solving
opportunities ask students to reproduce or minimally expand on things they have
already encountered. When opportunities do arise for inventiveness, they require
almost entirely equation-based modeling. These opportunities also typically come a
significant time after the toy model has been introduced. In the section below, I will
argue that adaptation of representational forms can be a means of learning. This
implies that instructors should consider a “learning-with” framing, an adapting-toy-
models framing, early on in the learning process.
Adaptation as a form of learning
Leveraging known models and representations, in their entirety or in a
piecewise manner, can be seen as a type of adaptive expertise. This type of sense-
making balances some dimensions of efficiency and innovation when modeling a new
situation (Schwartz, Bransford, & Sears, 2005; Rebello, 2009). An adaptive expert
flexibly utilizes aspects of known models that apply to the context in accordance with
local constraints to develop a runnable model (Rebello, 2009). Adaptive expertise is
characteristic of professional practice and is therefore an important target of
instruction (Hatano & Oura, 2003; Bransford & Schwartz, 1999).
6 https://depts.washington.edu/uwpeg/tutorials-QM 7 https://www.physport.org/examples/quilts/index.cfm 8 http://umaine.edu/per/projects/iqp/
96
Schwartz, Bransford, a Sears (2005) argue that there may be some “hidden
efficacy” in giving students early opportunities for adaptation and invention. Such
opportunities better position students to learn later on. To support this, they cite two
of their previous studies utilizing double-transfer experiments (Schwartz &
Bransford, 1998; Schwartz & Martin, 2004). In these studies, students are essentially
divided into two conditions. One condition has student invent a method of data
analysis. In the other condition students are given and practice a canonical method of
analysis. Both groups then receive a common resource to learn from and are assessed.
In both studies, students who were in the invent-a-method condition performed
significantly better than students in the tell-and-practice condition. They cite some
perceptual learning literature as providing a possible explanation for these results.
Analyzing the two cases side-by-side allows learners to perceive differentiating
features of the cases (Gibson & Gibson, 1955). Attention to these features may then
help guide student learning from more traditional resources like book chapters or
lectures.
I see using these toy models of QM as a particularly rich site for helping
students develop the professional vision I associate with seeing the world from not
only a physics perspective, but a physics perspective that deals with especially small-
scale phenomena. These models become a means of highlighting and representing
situations to make mechanistic, causal relations about particular entities more
apparent. In this view, learning with and about a QM toy model or iconic
representation entails more than simply learning how to reproduce and/or point to
97
important features/aspects of a representations. Learning here entails using the toy
model structure to change the way one sees new situations and interacts with the
world while “grappling with the core ideas of a discipline,” (Rosebery, Ogonowski,
DiSchino, and Warren, 2010).
Current physics education research has underexplored student reasoning in the
context of adapting toy models and iconic representations in QM to novel situations
There has been some focus in physics education research on student
understanding of toy QM models and their iconic graphical representations. The focus
tends to be on what students have learned (or not) about the particular models (Bao,
199; Sadaghiani, 2005; Steinberg, Oberem, & McDermott, 1996; Muller & Wiesner,
2002; Olsen, 2002, Domert, Linder, & Ingerman, 2004; Ambrose, 1999; Singh and
Marshman, 2015; Singh, 2008). There is a smaller portion of the literature that
touches on what ways, if at all, toy models can be tools for students to learn with
(McKagan et al, 2010; Cataloglu and Robinett, 2002; Morgan, Wittmann, &
Thompson, 2003; Petri & Niedderer, 1998). However, the focus of half of these
works is still mainly understanding student difficulties (McKagan et al, 2010;
Cataloglu and Robinett, 2002; Morgan, Wittmann, & Thompson, 2003)
In introductory quantum mechanics, iconic representations are an early target
of expertise for students. Traditional instruction tends to focus heavily on these
models and representations early on. This seems designed to get students to master
the fundamentals of these representations and models before moving on to more
98
complex systems like the hydrogen atom (Griffiths, 2016; Serway, Moses, & Moyer,
2004; Liboff, 2003). Accordingly, there has been a good deal of research and
curriculum development around student use and understanding of various iconic
representations (Bao, 199; Sadaghiani, 2005; Steinberg, Oberem, & McDermott,
1996; Muller & Wiesner, 2002; Olsen, 2002, Domert, Linder, & Ingerman, 2004;
Ambrose, 1999; Singh and Marshman, 2015; Singh, 2008). This work tends to
catalog student difficulties observed by researchers.
The collection of research on student difficulties also documents difficulties
with classical or statistical physics that may influence student understanding of
quantum mechanics, such as difficulties with probability (Sadaghiani, Bao, 2006;
Bao, Redish, 2001; Domert, Linder, & Ingerman, 2004) or the wave model of light
(Steinberg, Wittmann, Bao, Redish, 1999). This work indicates that some difficulties
in QM may result from a lack of understanding of more classical models or notions,
such as probability. Implicitly, this work suggests that students need to master the
fundamental concepts and the toy models before engaging in adaptation.
While there is a large body of research seeking to understand how students
deal with the toy models and iconic representations themselves, a smaller group have
started to investigate whether, and how, students adapt what they have learned about
iconic representations and toy models to more distant contexts. This work includes
the development of the Quantum Mechanics Visualization Instrument, which is
designed to test student ability to reason beyond more traditional quantum mechanics
problems (Cataloglu and Robinett, 2002). With several courses having taken the
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inventory, Cataloglu and Robinett found that students seemed reasonably capable of
generalizing the 1D infinite well to 2D. In generalizing, students were able to
correctly identify the algebraic and graphical forms of the energy eigenstates.
However, students seem less consistent in generalizing wavefunction forms of the 1D
infinite well to a spherical potential. Similarly, McKagan et al. (2010) in their design
and validation of the Quantum Mechanics Conceptual Survey, McKagan et al. report
on a collection of responses to a problem asking students to reason about a higher
order wave function in a slanted well9. In a course where the instructor had a
particular focus on discussing how the kinetic energy “encodes the curvature” of the
wavefunction, the students were better able to reason about the correct shape of the
wavefunction (McKagan et al, 2010). Also, McKagan and colleagues noted that
students who answered this question correctly were transferring knowledge to the
situation. In particular, one student transferred representational knowledge about the
wavefunction of a particle in a step potential. Additionally, Morgan, Wittmann, and
Thompson (2003) found that a small sample of students were able to correctly reason
about the shapes of wavefunctions in novel contexts. These students adapted
sinusoidal and decay pieces depending on the relative value of the particle’s energy
and potential in the region of interest.
The work reviewed above (McKagan et al, 2010; Cataloglu and Robinett,
2002; Morgan, Wittmann, and Thompson, 2003) shows that students can take aspects
9 See below for a description of the problem.
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of iconic representations from toy models to represent different situations. What these
works do is provide evidence of the “end products” of such application. However,
these works fail to provide any fine-grained account of how toy models can be tools
for students modeling these new situations. Such accounts require analysis of the
process of sense-making, not just the outcome. Our work will help show some of the
ways students tend to adapt the toy models in order to help complement existing
literature and to make a case to instructors for adaptation early on in a course.
It may seem that, given the wealth of research showing students have
difficulties with fundamental quantum concepts and representations, instruction
should focus on these first. However, it may be that the research community’s major
focus on student difficulties overly colors our perception of what students are capable
of doing and where instructors should focus their attention and resources. I hope to
add to work by others in the field, who are going beyond a focus on student
difficulties (e.g. see Dosa & Russ, 2016; Hammer, 2000; Smith, Disessa, &
Roschelle, 1994).
A similar issue arose in mathematics research education. Early studies of
students’ algebraic reasoning focused strongly on difficulties (Schliemann, 2007).
This reinforced a curricula set-up that kept algebra later in the curriculum and
introduced transitional courses like pre-algebra, all towards making sure algebra came
at a time when students were ready and able. Proponents of early algebra argue that
“a deep understanding of arithmetic requires certain mathematical generalizations,”
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which may be afforded through algebraic notation (Schliemann, 2007 citing:
Brizuela, 2004; Carraher, Schliemann, & Brizuela, 1999, 2000).
Methods (Analytical Flow)
How we created opportunities for adaptation of toy models and iconic representations
I wanted to create opportunities for adaptation to study how students use toy
models as tools to learn with. Students learn to construct and interpret representations
in disciplinary authentic ways by engaging in the practices of sense-making and
communication involving representations (Greeno & Hall, 1997). These include
discussions around conventions of interpretation of toy models, the affordances and
constraints of disciplinary representational norms. Engaging students in these types of
discussions in the context of content questions helps reveal different ways of talking,
thinking, and reasoning that are well-grounded in problem-solving experiences. The
goal of our interview design was to create situations where I might be able to study
whether and how toy models become tools for learning with.
These two principles helped guide task design:1) Continuity in content: The
researchers saw continuity between the new situations being modeled and various toy
models. Toy model prompts are immediately preceding the prompts designed to
promote adaptation to help prime or cue those models.
2) Interactional space: The space intended to be filled with a lot of student
talk, moves by the interviewer are meant to make student thinking more visible. In
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reasoning out loud with the interviewer students are also justifying their constructions
against disciplinary standards and norms.
Data Context
My data set was selected from a collection of 14 hour-long interviews with a
mix of physics and engineering students. Most students were of junior or senior
standing. In the interviews, the students were given tutorial-style problem sets and
were asked to think aloud with the interviewer as they went through the problem sets.
If there were moments of silence, the interviewer would check in. Questions directed
to students were meant to make the students' thinking more explicit.
The interviews were taken in two sets, with similar interview protocols for
each set. On the first set of interviews, the protocol included a problem on an infinite
well with a slanted problem, which I refer to as the “slanted well” problem (described
in more detail below) (McKagan et al., 2010, Cataloglu & Robinett, 2002). When
viewing these interviews, I noticed a relatively consistent pattern of students finding
ways to adapt the representation of the infinite square well to reason about the shape
of the slanted well wavefunction. I then designed an interview protocol explicitly
around creating opportunities for this type of sense-making. In the design of the
second interview protocol, students first encountered problems on the particle in a
box; these included drawing energy eigenstates, describing the energies, and thinking
about the speed of the particle. The next problem is on a classical analog of the
particle in a box, the “classical well” problem (described in detail below). This is
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followed by several problems on explaining quantum systems to peers, the slanted
well problem, and several problems on tunneling. Evidence of students adapting
iconic representations occurred on the classical and slanted well problems. However,
not all students were able to get to these questions in the protocol. Across both
interview sets, I had a total of 17 episodes of students reasoning about the slanted and
classical well problems.
Classical Well Problem
Problem statement: Suppose you had a classical particle in a physical situation
analogous to the quantum particle in the box. Consider a bead on a string, and the
string is knotted at x=0 and x=L so that the bead is confined between 0 to L, and can
move smoothly and freely between these bounds. The bead has some energy E, and
can bounce elastically at the knotted ends. Sketch the wavefunction of this classical
particle.
Solution: An anticipated solution would be a flat wavefunction, as the particle spends
equal time between the knots and can therefore be found at each position with equal
likelihood. Some students attend to the behavior at the edges, either suggesting the
bead cannot be found on the knot or modeling the acceleration of the bead at the
edges.
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Slanted Well Problem
Problem statement: Consider a quantum system with V(x) = ∞ for x=(-∞, 0) and
(L, ∞), and V(x)= Ax for x=(0, L). Sketch the wavefunction for the first allowed state,
or ground state, of the particle.
Solution: The canonical solution for the ground state is a first-order Airy function.
On this problem, I only expected students to reason about the shape of the
wavefunction.
Identifying Relevant Case studies for Analysis
The goal of my analysis is to understand how toy model representations are
used to make sense of new contexts. Previous work has suggested that adaptation can
happen (McKagan et al, 2010; Cataloglu and Robinett, 2002; Morgan, Wittmann, and
Thompson, 2003), but does not elucidate the process and mechanism behind such
change. Here, the orientation of microgenetic analysis provides guidance in studying
processes of change (Siegler &Crowley, 1991). In providing accounts of cognitive
development, 1) observations should occur throughout the process, 2) the density of
observations should be large with respect to the rate of change, 3) trial-by-trial
analyses should be used to model mechanism (Siegler & Crowley, 1991). Video data
allows such close, detailed observation. It also enables the data to be viewed by many
people across time, thereby allowing trial-by-trial analyses to develop through
repeated viewing of the data.
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My analysis begins with preliminary clipping of the video in order to hone in
on places where I think I see toy model representations (and associated concepts)
being applied to new contexts. This preliminary analysis included content logging the
interviews for general observations and moving between collective and individual
viewing of the data to reach some consensus on there is evidence of reasoning with
and about toy model representations in the highlighted sections of the data (Jordan &
Henderson, 1995; Derry, Pea, Barron, Engle, Erickson, & Sherin, 2010).
Analytical Methods
Orientation
I closely study interaction to understand knowledge-in-use because I see
cognition and learning as inherently embedded in interaction with socio-material
settings. My work is guided by others who do fine-grained analyses of the assembly
of representational states through the coordination of different semiotic channels
(diSessa, 1991; Stevens and Hall, 1998). In studying this assembly, I aim to
understand how speech, gesture, and material structure are coordinated to generate a
representational state. This includes understanding how coordination evolves
throughout the interaction, i.e. how the meaning/utility of a representation may be
changing.
Researchers studying knowledge-in-use in interaction between people and
their settings have taken different views of the ontology of knowledge. The studies
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mentioned above study knowledge-in-use, but take different ontologies of knowledge
and units of analysis. For example, diSessa (1991) uses individual, internal cognitive
processes as units of analysis, whereas Stevens and Hall (1998) more so consider the
group interaction as a unit. Unlike diSessa, I refrain from doing a full-scale
knowledge analysis of the interaction. Instead, I use tools from interaction analysis
(described in more detail below), and my membership in the physics community, to
look at an interaction and describe the informational content of the interaction
(Hutchins, 2000). Because of this orientation, my analysis describes internal
transformations only when such transformations are not explainable from looking at
the details of interaction.
Tools from Conversation Analysis
Tools from conversation analysis allow one to frame an interaction as a sequence
of organized events, where one event follows sequentially from the previous (Sidnell,
2011). The aim of this analysis is then to describe how each event is shaped by the
previous event, and how the current shapes the next, and so forth. Tools from
interaction analysis allow me to categorize and make meaning from the organization
of a conversation. It also allows me to identify entities helping organize the
interaction and understand how representations become imbued with conceptual
meaning. For example, the coordination of saying “this wall” while tracing a potential
wall in a representation, reveals a potential wall construct as a relevant entity.
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Below, I list some of the ways that people coordinate speech, gesture, and material
resources:
Gesture; all forms of gesture can add semantic meaning to interaction.
o Environmentally-coupled; gestures that mark something in the
speaker’s environment. This may be in the form of tracing a drawing
or pointing to something.
o Discourse-coupled; gesture that does not clearly mark anything in the
speaker’s environment but unfolds with speech. i.e. Shrugging
shoulders when saying “I don’t know.”
o Stand-alone; gesture that does not pair with speech. i.e. Shrugging
shoulders
Organization and structure of speech; these tools help me understand what
meaning is being generated between participants through how the
conversation it structured. This includes understanding how particular ideas in
speech are meaningfully connected to each other (Sidnell, 2010; Schegloff,
2007).
o Turn-taking; individual turns at conversation
o Repair; attempts to alleviate conversational trouble or breakdowns in
mutual understanding
o turn construction; conversational turns are structurally comprised of
turn-construction units, which may be single words, clauses, questions,
etc.
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o adjacency pairs (Sacks and Schegloff, 1973); distributed
conversational sequence of two utterances, where the first-pair part
mutually constrains second-pair part
o Preference; some second-pair parts are organizationally “preferred” in
the sense that some second-pair parts make more significant progress
towards the joint enterprise underway
o Progression; on a larger scale than preference, there is a sense that the
conversation should move towards accomplishing the mutually
determined purpose
o Discourse markers (Schiffrin, 1988; Bolden, 2009); words or phrases
that help organize conversation into segments, and suggest meaningful
connection between those segments.
Example Analysis: interview with Quinn on the classical well problem
I present a short piece of transcript to provide an example of how I am
analyzing the data. The table 3.1 is split into three columns. The first column gives
the transcript. The second column describes what aspects of the transcript I am
attending too. The third column shows conclusions I make from what I notice in the
data. In the snippet, Quinn is working on the classical well problem. Immediately
before the transcript shown below, Quinn was reasoning about the system as if the
bead and string are both free to move. The interviewer then suggests adjusting the
situation being considered to one where the string is held taut.
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FIG. 3.1 Quinn’s Bead representation: Quinn’s drawing of the bead on a string.
Arrows show that the bead can move back and forth along the string.
Transcript Noticing → Meaning Analysis
QUINN: Ok, if that's
the case like ((draws
bead
representation)), so
it's just moving back
and forth? ((gestures
motion of the bead
over the
representation,
within bounds shown
in the
representation))
Ok (Schiffrin, 1988) →
discourse marker as check on
shared understanding
“If that’s the case” →
conditional statement
following a reframing move
from the interviewer
So → marks another turn in
the conversation (Bolden,
2009), showing conditional
Quinn first generates a
representation of the bead
(see Fig. 3.1), with which she
physically represents the
motion through her speech
“it’s just moving back and
forth” and simultaneous
environmentally-coupled
gesture. Given that this
interaction between Quinn
and the material structure she
has created occurs
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statement not completed
It’s just moving back and
forth? ((gestures over bead
representation in Fig. 3.1)) →
Coordination between
gesture, speech and
representation creates an
embodied representation of
the bead’s motion. We need
all channels to get the fullest
understanding of the motion;
the gesture shows smooth,
linear motion that’s bounded
“by” the edges of the string
marked in the representation.
immediately after the
interviewer has reframed the
situation being discussed,
Quinn prefaces her statement
with “if that’s the case”, and
that she ends her
conversational turn with a
question, this suggests that
Quinn is using her turn to
check in to see if she is
conceptualizing the motion of
the bead correctly. So, not
only is Quinn coordinating
material, embodied, and
discursive (“moving back and
forth”) to generate a
representation of the bead;
she is also generating a
representation in a way that is
apparent to the interviewer, as
a means of checking for
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mutual understanding.
INTERVIEWER:
Yeah, bouncing at
the knotted ends
((gestures back and
forth motion over
table)).
Completion of question-
answer pair → affirmative to
Quinn
Interviewer utilizes speech
and gesture to generate her
own, similar representation of
the bead’s motion following
question-answer
Following Quinn’s attempt to
reach a shared understanding
of the bead’s motion, the
interviewer starts her turn
with an affirmation. She then
includes a rephrasing of the
description of the motion
(“bouncing”) along with an
extension of the description
(the motion turns back at the
“knotted ends” with a
simultaneously gesture,
which closely mirrors that of
Quinn in the previous turn.
The boundedness of the
motion was previously
perceptually available in the
interaction in that Quinn’s
embodied representation of
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the motion ended at the
knotted ends. This notion
becomes discursively
available as the interviewer
both notes this verbally, and
in her gesturing of the beads
motion, which shows the
motion punctuated at
particular points. The
affirmation at the beginning
of the turn, the rephrasing and
extension of the
representation of the bead’s
motion, along with the back-
and-forth gesture being used
by both participants suggests
the two have reached a shared
understanding of the bead’s
motion.
QUINN: Oh, ok. So, Oh → discourse marker Quinn’s opening statement of
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I think we're still
gonna have the same
condition as that one
((gestures to
previous prompts10)).
marking receipt of
information
Ok→ discourse marker as
reference to action of the base
pair (question-answer). In this
context, it seems to register
acknowledgment of shared
understanding of the bead’s
motion.
So → discourse marker
showing turn in the
conversation following
acknowledgement of shared
understanding of motion. The
connection also seems to
imply some causal linkage
between motion and
“condition” i.e. motion
implies condition
“oh, ok. So,” followed by a
focal turn in conversation,
moving from talking about
the motion of the bead to
“conditions,” suggests Quinn
sees a shared understanding
of the situation. Her follow-
up, saying the “condition” of
one system is the same as
what she experienced while
working previously. Context
of the prompts (previous and
current) suggest that she is
referring to the wavefunction
as the “condition.”
10
Previous prompts were questions on the infinite well.
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INTERVIEWER:
Which one?
QUINN: As the, the
particle.
INTERVIEWER:
The infinite box?
QUINN: Yeah. The
infinite box
((gestures to
previous prompts)).
Question-answer
Question - answer →
Repetition of the question
implies a non-satisfactory
answer with “the particle” …
asking about what is the
“one”
Environmentally-coupled
gesture coordinated with “the
infinite box”, follow-up turn
unit used to the answer in the
question-answer pair is a
rephrasing or renaming
(shows continued search for
shared understanding of
referential system)
In her previous turn, Quinn
elucidated “that one” through
gesture to previous prompts.
However, the interviews next
turn, “which one?” (Emphasis
added), shows to which
system Quinn is referring is
unclear to the interviewer.
Quinn’s response is not
enough to make the
clarification for the
interviewer, as when Quinn
says “the particle,” the
interviewer again asks a
clarifying question, asking
whether it is the “infinite
box” that she is referring to.
Quinn starts her turn with an
affirmative “yeah,” then takes
on the interviewer's language
to reference the particular
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system “infinite box” as she
again gestures to the previous
prompts. These coupled
actions show that Quinn and
the interview share a sense of
which system she is
borrowing a wavefunction
(“condition”) from the
infinite box.
Table 3.1: Example analysis of episode in interview with Quinn
1. The example above showcases how I am engaging in a microgenetic analysis
and in doing so draw on multimodal analysis and tools from conversation and
interaction analysis.
2. In the story so far, Quinn has drawn on the infinite well toy model and
representation to reason about the bead-on-a-string. In her reasoning, she has
drawn on the infinite well to model the movement of the bead and the string.
3. I pick up the story below in the Results section. In the next snippet, the
interviewer reframes the situation to considering to one in which the string is
held taut.
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Results
Preview
I find that students frequently do see continuities (and discontinuities) across
situations involving toy models. The different continuities that individual students
notice help shape the way the student sees, highlights, represents, and reasons in the
new situation. I try to differentiate these interactions taken by students some by
pointing out what aspects of the iconic representational form are used in modeling the
new situation. In some cases, students may see the entire physical form (with some
conceptual meaning layered) as applicable, in other cases students may see smaller,
modular portions of the form being applicable. For example, in looking at a problem
that involves some tunneling, a student may reference the finite well representation,
repurposing the representational form and conceptual notion of decay. In other cases,
students may see the entire form as applying with some transformation of that form,
or pieces within. For example, in borrowing the decay form from the finite well, a
student may note that the piece actually needs to “decay sharper” than what’s
experienced in the finite well situation because the potential of the current problem
grows, as opposed to staying constant.
My analyses show that students drag pieces of toy model representations to
incorporate into new representations. This process helps shape and focus students’
sense-making. In some cases, the iconic representation provides a pathway to
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comparing and contrasting fundamental behavior. In my analyses, I show that
learning with a toy QM model often provides a means of focusing sense-making to
particular regions in the toy model (and referent physical set-up) or regions that
emerge from sense-making.
Quinn on the classical well problem (continued)
Infinite well representation as a first model for the bead-on-a-string
In the illustrated analysis in table 3.1, I showed how from the outset, Quinn
sees continuities between the infinite well and the bead-on-a-string. I now pick up
this story in this section. The interviewer now attempts to reframe Quinn’s
understanding of the set-up by suggesting that she consider a situation in which
the string cannot move. In considering this new situation, Quinn still sees
continuities between the infinite well and the bead-on-a-string. Seeing the bead
through the lens of the infinite well directs Quinn’s sense-making to the center of
the bead’s motion and the center of the infinite well representation.
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Fig 3.2. Quinn’s Bead representation: Quinn’s drawing of the bead on a string.
Arrows show that the bead can move back and forth along the string.
QUINN: Oh ok. So, I think we're still gonna have the same
condition as that one ((gestures to previous prompts)).
INTERVIEWER: Which one?
QUINN: As the, the particle.
INTERVIEWER: The infinite box?
QUINN: Yeah. The infinite box ((gestures to previous prompts)).
Because, so this is like the x-axis right? ((Gestures to
horizontal axis in bead representation)). And the
particle is moving, and we're seeing where it's gonna be
((gestures back and forth over bead representation)).
So, when it moves this way and comes back ((gestures
back and forth over bead representation)), it will reach
the middle one twice ((points to center of bead
representation)). So, there is a bigger probability of
seeing the particle there ((points to center of bead
representation)). So, in that sense it might look like this,
this being the x-axis and this being the probability
((draws probability representation)).
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FIG 3.3. Quinn’s Probability Representation: Quinn’s representation of probability
versus position of the bead-on-a-string.
Quinn uses the infinite well representation as a means of representing and
sense-making around the bead-on-a-string. In doing so, she reasons about the most
likely location of the quantum particle and the bead.
Seeing the bead-on-a-string through the lens of the infinite well helps direct
Quinn’s sense-making. Quinn reasons only about the center point of the infinite well
wavefunction and the bead’s motion as a means of inferring a wavefunction for the
classical bead. Quinn describes that “we’re seeing where it’s gonna be” as she
represents the motion of the bead with her pencil. With joint attention on the
represented motion of the bead, Quinn highlights the center point of the motion. She
then infers that the probability will be larger at this point because this point gets “hit
twice.” She moves down the page and draws a representation of the probability versus
x-axis. The wavefunction she draws is also the ground state of the infinite well.
Through her highly localized sense-making, Quinn develops a sense that both
the classical and quantum particles have the same probability distributions. However,
these two systems should not have the same probability distributions. At this point,
Quinn’s drawing on the infinite well may look unproductive towards generating a
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correct solution for the bead-on-a-string. In the coming sections, slight shifts in
Quinn’s perceptual attention to the situation helps Quinn develop a deeper
understanding of both the infinite well particle and the bead-on-a-string. This deeper
understanding very much relies on Quinn having taken this “incorrect” step in her
modeling.
In the next sub-section below, I continue the story of Quinn’s sense-making
about the bead-on-a-string.
Intervention: Attention to new regions shifts Quinn’s modeling of the bead-on-a-
string
In the previous section, Quinn had only reasoned about the center point of the
bead’s motion and the infinite well wavefunction. Next, Quinn broadens her attention
and sense-making to consider new regions in both representations. Doing so helps
Quinn develop a deeper understanding of the infinite well wavefunction and the bead-
on-a-string.
INTERVIEWER: Can I ask you if I have two points like here and here
((points to two points on probability representation)).
QUINN: Two points here and here ((points to two points above curve in
probability representation))?
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INTERVIEWER: Along the curve ((points to two points on probability
representation)), um and that corresponds to two points here and here ((points
to two points on bead representation)). This tells me that this point is more
likely than that point ((points to two points on probability representation)).
QUINN: Ye:ah.
INTERVIEWER: Is that right?
QUINN: No.
INTERVIEWER: Why do you say it's not right?
QUINN: Yeah. Because it's gonna be uniform. The particle, it’s the same
particle, and it's moving this way, you can see it the all the way and coming
back ((traces over probability representation)). So it's definite that it's going to
reach this point twice and this point twice ((points to two points on probability
representation)). And I think the probability will be uniform.
INTERVIEWER: So what would the wavefunction look like do you think?
QUINN: In that sense, I think it would just be like ((draws flat line
wavefunction in refined probability representation))
INTERVIEWER: A flat bar?
QUINN: Yeah.
INTERVIEWER: So everywhere is be equally likely?
QUINN: Mhm.
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FIG 3.4. Quinn’s Refined probability Representation: Quinn’s representation of
probability versus position of the bead-on-a-string.
Attention to new regions in her two representations, and conceptual
coordination of those regions, helps Quinn deepen her understanding of the two
systems and their probability distributions.
Quinn’s speech shows that she sees the differential likelihood encoded in the
infinite well representation and that the differential likelihood of those two locations
may not apply here (“ye:ah”). Quinn essentially repeats her sense-making around the
motion of the bead. In doing so, she again simulates the motion of the bead and reads-
out where the particle hits twice. This time, however, her sense-making also attends
to the two points of the infinite well and the bead’s motion. Like the center point in
the previous simulation, these points are also reasoned as getting being reached twice.
Though her sense-making is not substantially different, directing her reasoning to
these new regions in her representations leads to a new understanding of the bead’s
physical behavior and its associated wavefunction. Quinn now concludes that the
probability of the bead should be uniform across the string.
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Her expanded sense-making leads her to realize that the bead-on-a-string and the
infinite wells have different probability distributions. The physics of the bead-on-a-
string and the quantum particle are very different. This is in sharp contrast with her
modeling of the situation, where she explicitly stated that the behavior of the two
systems is the same. The question is, will she notice this foundational concept that has
emerged from her own sense-making?
Reflections on the differences in ontology of the infinite well particle and the
bead-on-a-string
Quinn does notice the profound consequences of her reasoning. Her reflection on
the differences in the wavefunctions leads to an insight into the ontological
differences of the two entities.
QUINN: I'm wondering why the particle in quantum mechanics have this kind
of wavefunction, because-- yeah I guess it makes sense. Maybe it's because
it's not a particle particle, it's a wave. So it would disperse and change its
shape. Yeah but this one is a particle and if we see it like that, it has to look
like something flat.
Quinn’s rejection of the infinite well representation as also a representation
for the bead enabled her to reflect on the ontological distinctions of the quantum and
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classical particles. Careful ontological distinctions of entities is a crucial element of
developing expertise in quantum mechanics (Brookes & Etkina, 2007).
Discussion of Quinn
Recounting Quinn’s sense-making: how Quinn’s reasoning developed through
attention to new representational features
Quinn began the episode by immediately recruiting the infinite well
representation as a means of explaining the wavefunction for the classical bead. The
infinite well was a relatively stable means of seeing, representing, and sense-making
about the bead-on-a-string. As the episode progressed, Quinn reasoned about
additional features of the infinite well representation which led to additional sense-
making in new regions of the physical setup. Attention to new areas in both contexts
lead to a distinction between the wavefunction distributions of the classical and
quantum particles. Quinn ultimately found a limit for the infinite well model and
rejected the representation as appropriate.
Opportunities for gaining conceptual insight: 1) Coordinating sense-making
around new representational features, and 2) Deeper understanding of
ontological differences
I discussed two ways in which Quinn may have gained some conceptual
insight in this episode. First, in her reasoning, Quinn began to sensemake around
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additionally relevant features of the infinite well wavefunction. She went from
reasoning about the most likely location to differences in probability along the
horizontal axis. As she reasoned, new features of both spaces became relevant and
provided the mechanism for the eventual separation of the classical and quantum
models. Highlighting or attending to new visual features, and coordinating subsequent
action or sense-making around those features, is fundamental to learning (Parnafes,
2007; Goodwin, 1994).
Second, the prompt provided an opportunity for Quinn to reflect on the
ontological differences between quantum and classical particles. Both threads may
help Quinn develop a deeper understanding of the behavior of quantum particles.
Previous prompts in the interview were on the quantum infinite well. These included
questions asking students to draw the first few energy eigenstates and find the most
and least likely locations of the particle in the n=2 state. On the latter, Quinn correctly
drew the wavefunction and identified the least likely locations shown. However, she
said that “we might not find the electron there.” The interviewer asked if this meant
that it was impossible to find the electron there but she was unsure. It seems likely
that a developing intuition around the quantum particle as a wave may help Quinn
flesh out her uncertainty around the physical meaning of the nodes in the quantum
wavefunction.
Oliver on the classical well problem
Adding an offset to the infinite well representation to model the bead-on-a-string
126
Oliver begins by drawing a picture of the bead-on-a-string (fig. 3.5) and then
starts reasoning about the bead’s motion. In doing so, he compares the bead-on-a-
string to the infinite well. Discontinuities that Oliver perceives between the two
systems help shape Oliver’s sense-making of both systems.
FIG 3.5. Oliver’s Bead-on-a-string representation
After drawing a picture of the bead (fig. 3.5), Oliver reasons about the velocity of the
bead across the string:
OLIVER: It'll have a like, its velocity will peak in the middle and then be at a
minimum towards the edges ((vertical hands gesture edges)). And. That
technically is the same, I would say, for its position as well. Just because, it
wouldn't, I guess in this case though, you wouldn't have a probability of zero,
at the knots. Because uh, the bead will just bounce back off. Like you could
still catch the bead on the knot.
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Subsequent turns will reveal Oliver is making a comparison to the infinite
well (“in this case though”). It’s through the comparison to the infinite well that
Oliver moves from sense-making about position and velocity of the bead, to inferring
what this may mean for the wavefunction. His sense-making is focused on comparing
the edge-behavior of the two systems because this is where he sees a discontinuity
between the two systems.
Oliver says that “in this case though, you wouldn’t have a probability of zero,
at the knots.” The wavefunction in this case is non-zero, but the infinite well
wavefunction is zero at the edges. Oliver goes on to connect the physical behavior of
the bead to the nonzero probability. He notes that the “bead will just bounce back
off”, which for him implies that you can find the bead on the knot. In this way,
adapting the infinite well representation to the bead-on-a-string is focuses Oliver’s
physical sense-making about the two situations.
After reasoning about the bead’s position and velocity through comparison to
the infinite well, Oliver draws a wavefunction. He goes on to expand on the
discontinuity he perceives between the wavefunction for the classical and infinite
wells:
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FIG 3.6. Oliver’s Probability Representation: This is a recreation of Oliver’s first
attempt at a wavefunction for the bead-on-a-string. Later in the episode, his sense-
making leads him a new conclusion and so he erases his wavefunction in the graph
and redraws it. For this reason, I recreate his original drawing.
OLIVER: This is basically the wavefunction for the bead. And, the difference
that it has with, like a particle in an infinite well, is that it doesn't touch the
edges, like there's still a little bit of distance between the minimum values of
the wavefunction and I guess the zero of the, or just the ground level. Or V=0.
INTERVIEWER: So physically, what does that mean?
OLIVER: Physically, this just means that you can find the bead AT the knot.
As opp-, you can have the bead at the knot, as opposed to the quantum
particle, you can't really have the particle on the wall, because it will bounce
off.
129
Oliver continues his comparison to the infinite well, which also continues to
direct his sense-making to the edge behavior of the two systems. The comparison also
leads to sense-making about wavefunction representations more generally.
The wavefunction he has drawn is the ground state of the infinite well with an
offset, which emerges from being unable to find the quantum particle at the edge.
Oliver then reasons more deeply about the offset: What is the offset actually
measuring? The bead’s edge probability is nonzero, but nonzero with respect to what?
Here, Oliver makes several repairs while attempting to articulate what the distance
describes. However, uncertainty around how to describe the offset is not stopping
Oliver from engaging in some mechanistic reasoning around how the two entities are
behaving at their boundaries, and how that physical behavior leads to different
probability distributions. i.e. The difference in probability at the edges arises from
being about to find the bead, but not the quantum particle, at the knot. And so, even
though Oliver could be uncertain about how to describe the offset at the edge, this
does not stop him from reasoning about its physical significance.
Intervention: Attention to new regions leads to uncertainty in modeling the
bead-on-a-string as the infinite well with an offset
Oliver’s interview proceeded with roughly the same intervention as Quinn.
The interviewer first pointed out the differential likelihood of two off-center points
implied by Oliver’s representations and asked if that interpretation sounded right.
Oliver agreed and went on to explain why. Oliver reasons about the bead’s position to
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show why the differential likelihood applies to the bead. His reasoning leads him to
new understandings and new areas in his representations.
INTERVIEWER: So can I ask you why like, maybe two positions, here and
here, a little off center ((points to two points on wavefunction in probability
representation)), like here versus here ((points to two points in bead-on-a-
string representation)). Why, I think what this ((points to wavefunction in
probability representation)) shows me is like, this position ((points to point on
bead-on-a-string representation)) closer to the center, is more likely than
something like right next to it. Does that, it, does that sounds right?
OLIVER: Ye:ah. So, essentially, ummm. Similar to how, with a classical
particle, it bounces off the barrier, so to speak, it always hits, every point.
Well. Saying that you would travel, starting from zero ((left pointer into
table)) to L ((right hand into table)), and then back to zero ((left pointer into
table)). You're hitting every point twice ((points to space between where his
hands were)). So to speak. But. Hm. Actually. Hmmm, I don't know. Hmm.
INTERVIEWER: What are you thinking now?
OLIVER: Uhhhh, I'm not sure, well. In a sense, I guess I'm trying to, compare
this to, the quantum particle. But then again, this. The physics don't work the
same.
131
In response to the interviewer’s question, Oliver introduces a new line of
reasoning that also incorporates attention to new regions in the bead-on-a-string’s
motion to think about the differential likelihood in the infinite well. He gestures the
motion of the bead and reasons about what points get hit twice. This is in contrast to
Oliver’s first line of reasoning (momentum and velocity), where Oliver only reasoned
about the edges and center. His new line of reasoning is not differentiating the
different points along the string. Instead, his reasoning suggests that every point is the
same because every point gets hit twice. Oliver seems surprised by this new
understanding, suggesting that he may now be unsure of his earlier reasoning. At the
interviewer’s prompting, he steps back to express his lack of surety through
explaining what he’s trying to do: compare to the “quantum particle.” He goes on to
stress that there are fundamental differences in the physics of the two situations.
Oliver holds onto sinusoidal wavefunction for the bead-on-a-string even though
his reasoning does not back it up
Oliver continues reasoning about what points get hit twice. It’s clear that he
wants to hold onto the sinusoidal wavefunction shape but his developing reasoning,
which considers more areas of the bead’s motion, is leading to other conclusions.
OLIVER: Like it always ends up hitting the middle point twice ((pen into
table)). Or the middle points twice ((gestures range with both hands)), more
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than it hits the edges. But that still doesn't really, it doesn't really all the way
explain why, you're not, why the middle points aren't counted equally
((gestures range with hands)). So like why it does, why isn't it just a step
function as opposed to a sinusoid. Hmmm.
OLIVER: This is. Well. mv is the momentum. Its mass is the same. The only
thing that's changing is the velocity. And velocity in terms of x. So.
Oliver is now more explicit about how his reasoning about what points get hit
is yielding a deeper understanding of the bead’s motion that makes the infinite well’s
sinusoidal shape a now questionable model for the bead. His reasoning is leading to
the new conclusion that middle points should be counted equally. But he seems
reluctant to let his initial sinusoidal representation of the probability go.
Attention to new regions in velocity representation finally disrupts Oliver’s
initial modeling of the bead-on-a-string
Oliver has returned to his initial entry point to the problem, reasoning about
momentum and velocity of the bead. The interviewer asks for a representation of the
velocity so that they can better sensemake around it.
133
FIG 3.7. Oliver’s Velocity Representation: This is a recreation of Oliver’s first
attempt at graph for the velocity of the bead-on-a-string. His sense-making later in the
episode leads to a more nuanced understanding of the velocity and so he erases his
first velocity curve. I recreate his original.
Oliver draws the “first half of a sinusoid,” and again mentions that velocity
will “maximize” in the center. The interviewer then directs Oliver’s attention to half
of his velocity plot and draws some implications from Oliver’s reasoning and
representation. Doing so leads to a new understanding of the forces, acceleration, and
velocity of the bead, which helps him understand the probability of the bead to be
uniform.
INTERVIEWER: So can I ask, in the first, like half, what is causing the bead
to accelerate ((points to right half of velocity representation))? And then
decelerate?
OLIVER: Ummm. Hm
134
OLIVER: This is true. Technically, there isn't anything decelerating the bead.
Whatever initial force kind of, hits the bead, it'll shoot to the velocity it's
supposed to and then drop back off. ((erases velocity representation))
OLIVER: Which probably ends up meaning that, the:.... the same thing holds
for the wavefunction as well. Since, ((erases wavefunction in probability
representation)) the velocity is a constant, its position is also, going to be, well
it, the position, the position itself is not going to be constant. But, in terms of,
predicting where it's going to be, you know that it's just going to be
somewhere in between the two knots equally, somewhere between those knots
((draws flat wavefunction in his probability representation)).
Reasoning around half of the velocity plot leads to some consideration of the
relationship between forces, acceleration, and velocity of the system. Coordination of
these constructs across the string (“between those knots”) allows Oliver to conclude a
flat wavefunction for the bead.
At this point, I end my narration of the interview to discuss how Oliver’s story
connects to the main claims of this chapter.
135
FIG 3.8. Oliver’s Refined Probability and velocity Representations: Oliver’s refined
representations of probability versus position of the bead-on-a-string and velocity
versus position.
Discussion of Oliver
Adapting the infinite well toy model to the bead-on-a-string explicitly helped
shape Oliver’s sense-making around the bead-on-a-string and the infinite well. Oliver
saw a discontinuity between the two situations in terms of the probability of finding
the two different entities at their boundaries. This helped focus Oliver’s sense-making
to the edge behavior of both situations. Like Quinn, Oliver’s adaptation of the infinite
well representation to model the bead-on-a-string leads to an incorrect result, a
sinusoidal wavefunction as opposed to a flat wavefunction.
Also like Quinn, attention to new regions in his representations, and new
conceptual coordination around those regions, helps Oliver find his way to a correct
understanding of the probability distribution of the bead-on-a-string. Getting to this
136
understanding involved the necessary conceptual steps of 1) developing a deeper
understanding of the velocity of the bead, through sense-making about particular
regions of the bead’s movement, 2) sense-making around the differential likelihood of
the probability distribution of the infinite well particle.
Chad on the slanted well problem
Finite well: a representational space for reasoning about how potential walls
affect the wavefunction through “decay”
I now turn to Chad, specifically the place in the interview where Chad and Erin
are discussing the slanted well problem. After reading the problem, Chad begins by
noting that the slanted well was a problem on a final exam, although it was not
something they had talked about in class. However, in the space of the interview, we
do not see Chad reproducing his work from the final exam. Instead what we see is in-
the-moment sense-making about the slanted well as Chad assembles a solution from
the coordination of two toy model representations.
Chad draws on the finite well representation to explain how the walls of the
potential well will have an effect on the wavefunction of the slanted well. In doing so,
Chad focuses his sense-making on how potential walls and regions shape the
wavefunction of the finite well.
Chad: We talk about how the potential walls affect it ((traces vertical wall on
slanted well representation)). And how it would be uhh... ((re-traces vertical
137
wall on slanted well representation)) Yeah 'cus we talked, yeah if you talk
about, uhh finite regions ((starts drawing finite well representation)), you have
the wavefunction in here, it doesn't go to zero here ((starts drawing
wavefunction in finite well representation)), it goes to the points that it does,
then exponentially decays in it ((finishes drawing wavefunction in finite well
representation)).
FIG 3.9. Chad’s Slanted well representation: Chad first draws the potential of the
slanted well.
FIG 3.10. Chad’s Finite well representation: Chad’s completed drawing of the finite
well representation
Chad begins with the claim that the “potential walls affect it.” However, his
explanation stalls after this point, with three clauses showing failed attempts to
138
continue justifying how potential walls affect the wavefunction. With the generation
of the finite well representation, he is able to continue. In doing so, he talks about the
wavefunction in the finite well and focuses on the behavior in the classically-
forbidden region. He describes the behavior as “exponentially decay[ing] in it.” He
also describes the values at the boundaries, which are non-zero at the inner boundary
(above), and “if they’re tall enough, you get tunneling,” (subsequent conversational
turn).
Chad goes on to draw the wavefunction for the slanted well, and in doing so,
recruits the conceptual and representational notion of decay from the finite well to
describe the right side of the slanted well wavefunction.
Chad: For this one it would just be, uhh it starts off like it, and then it decays
((draws ground state as he speaks)).
139
FIG. 3.11. Chad’s Slanted well representation: Chad draws the wavefunction in the
slanted well potential.
Recruiting the finite well representation helps Chad reason about potential wall’s
effect on the wavefunction. Sense-making around different regions in the finite well
then helps shape Chad’s compartmentalized sense-making around the slanted well
wavefunction. Chad has essentially partitioned the construction of the wavefunction
into two halves, where the halves are being explicitly recruited from other systems.
The right half of the wavefunction is taken from the classically-forbidden region of
the finite well representation. In Chad’s next turn, he makes it clear that the left-half
of his construction is recruited from the infinite well representation.
Infinite well: a representational space to further explore how potential walls
affect the wavefunction
140
Chad goes over his construction of the slanted well wavefunction. This time, he
draws on the infinite well representation to reason further about how the potential
walls affect the wavefunction. Chad’s deepening reasoning leads to the emergence of
a new conceptual and representational feature in his wavefunction.
Chad: So it's, you can kind of take it as perturbation upon the particle in the
box ((begins to draw infinite well representation)). So it's going to be
essentially particle in the box ((draws n=1,2 in infinite well representation))
but then, uhh what's happening is as the potential increases ((draws potential
slant)), it reduces the probability of being in that region. Which means that if
you still normalize it ((traces n=2 in infinite well representation)), it would
have to follow, it would have to follow the same energy, stepping, where it's
going by nodes added, but it will reduce the probability of this region ((circles
right hand side of n=2 in infinite well representation)), linearly.
Interviewer: Uhh, this region? Is that... ((Interviewer points to right side of
n=2 in slanted well representation))
Chad: Yeah, this is the ((shades under slant in slanted well representation))...
but, so it will follow essentially it ((traces left half of ground state of infinite
well in slanted well representation, leaving small line showing where two
wavefunctions deviate)).... I think I made it too big for my waves to look
right. But it will go into it generally like that ((points to n=1 in infinite well
141
representation)). But it will also decay ((traces remainder of ground state in
slanted well representation)) after it enters the region.
FIG 3.12. Chad’s Infinite well representation above the slanted potential
representation: Chad draws diagonal line (slanted potential representation) directly
below his infinite well representation. Lettering below the slanted well representation
occurred later in the episode and so should be ignored.
142
FIG 3.13. Chad’s Slanted well representation: Chad has traced over his wavefunction
for the slanted well. In tracing over, Chad marks a mathematical/representational
turning point; the point where the wavefunction of the particle begins to deviate
noticeably from the wavefunction of the infinite well.
Through the introduction of the infinite well, Chad can reason more deeply
about the slanted well wavefunction and the relationship between the probability and
potential. In the first section, the slanted well wavefunction was treated as the
juxtaposition of the left half and the decay portion. With Chad’s perceptual and
conceptual coordination between the infinite well representation and potential slant
representation, Chad relays a more nuanced understanding of the probability and
potential relationship; the probability and potential being inversely related. This also
reveals a more nuanced understanding of the slanted well wavefunction. As opposed
to a simple juxtaposition, his explanation now positions the slanted well as one
influence (potential slant) modulating the other (infinite well). This in turn gives rise
to a new conceptual and representational feature of interest: a representational
143
turning-point where the modulation from the rising potential causes a noticeable
deviation of the slanted well wavefunction from the infinite well wavefunction.
Discussion of Chad
The introduction of the finite well helped focus Chad’s sense-making through
the coordination of his reasoning on how potential walls and regions affect the
wavefunction. To do so, Chad reasoned about different regions in the finite well
representation, focusing on how the values of the potential at walls or within regions
affected the wavefunction. And so, Chad’s adaptation of the finite well led him to
sense-make about it in this particular way.
Coordinating his reasoning with the infinite well helped Chad flesh out his
slanted well wavefunction model as two competing influences. This led to the
emergence of a turning-point marking where the decay influence ‘wins’ over the
infinite well influence.
Conclusions
Seeing continuities across situations
With this paper, we hoped to show that toy models can be tools to learn with,
even if the student has not developed a sophisticated understanding of the model a
priori. Above, we described in detail how each episode of adaptation was also a
learning process or presented rich opportunities for learning. In this section, we will
expand on these claims.
144
Seeing toy models as tools for learning with means first seeing continuity
across situations. However, in this data set, continuity was seen across various grain
sizes: from seeing the entire toy model representation as applicable to seeing smaller
“chunks” of the representation as applicable. After recruiting a toy model
representation to reason with, sometimes fleshing out specific differences between the
scenario of the iconic representation and the new application scenario can be helpful
in the generation of new meaning, as we saw in the case of Oliver. For example,
Oliver saw discontinuity between the quantum and classical infinite well situations in
the edge behavior of the particles and their associated representational forms. For
each student, continuities and differences helped shape and direct the students’ sense-
making.
First instincts
Across our data, we find students’ first, intuitive modeling of situations often
seems to come in terms of the toy models. We say it’s intuitive because we often
don’t know why the student is drawing on the toy model, and the comparison comes
almost immediately after the student reads the prompt. Attention to new areas in the
representations students draw or new areas in the referent physical system, and
conceptual coordination around the areas, then helps the students develop their sense-
making through the generation of new meaning. Sometimes the new meaning is
finding some conceptual and/or representational means for rejecting or refining the
initial, intuitive model. The fact that many of these quick responses come from toy
145
models is pretty significant. Yes, we designed for this. But this might be some hint
that toy models might be useful for having students develop intuition around the
behavior of quantum entities. Particularly through the refinement or development of
their initial instincts. For Quinn, the refinement of her initial orientation towards the
problem led to a reflection on very fundamental differences of quantum and classical
particles.
Sometimes first instincts towards a situation can be hard to disrupt. As we saw
with Oliver, it seemed that his intuition was drawing him towards sinusoidal shapes
for the velocity and wavefunction. It was through attention to, and conceptual
coordination of, new areas in the both the infinite well and the motion of the bead that
Oliver was finally able to appropriately refine his initial intuition. Being able to refine
a first instinct in favor of a model that makes more physical sense is an aspect of
adaptive expertise (Hatano & Inagaki, 1986).
Coherence seeking across situations
My claim in this chapter is that the process of adapting toy model
representations can naturally lead to opportunities to deepen one’s understanding of
the representation being adapted. In our description of Quinn’s sense-making around
the classical well problem above, we noted how Quinn’s episode ended on her
reflecting on the ontological difference between the quantum and classical particles.
In looking across the data set, we see that reflections on the adapted representation
occurs mainly on the classical well problem, suggesting the importance of task
146
structure. Of the four students who adapted the infinite well to reason about the
classical well particle, three of the students sought some sort of coherence between
the two systems. Two students found an explanation in a high-n limit, suggesting that
the classical particle might have a high-frequency wave function, (possibly due to
much higher energy), which the students both considered to be well-approximated by
a flat line. Oliver is the only student who moved on without reflecting on the two
systems. What’s interesting to note is that all four of these students began with a
wavefunction explicitly adapted from the infinite well but settled on a flat
wavefunction description. It seems that adapting the infinite well representation, and
then rejecting that adaptation, provided a nice material (for all students but Oliver, as
he erased the infinite well ground state) and temporal juxtaposition of the
representations for the two systems for the students to then compare.
Implications
Opportunities for adaptation
Something I have argued for in this chapter is seeing these QM toy models
and their representations as means for structuring one’s seeing and reasoning in new
contexts. This is true even for students who have not yet developed a sophisticated
understanding of the toy model. In doing so, I want to help push back against the
notion of waiting until students acquire some pre-determined, culturally-sanctioned
representational and conceptual understanding before allowing students to reason
about more complex situations. Understanding experience (in education and
147
otherwise) to be continuous, experiences with these toy models come to contextualize
current experiences for these students. Having this type of reasoning become
expected is then a matter of designing learning experiences so that students can make
more meaningful connections, as well as opportunities to investigate and reflect on
those connections. As researchers, I help create situations that afford continuity
through interaction with the student and careful task design.
What if adaptation leads to incorrect results?
One concern for instructors may be that adaptation of toy models can lead to
incorrect results. Additionally, it’s difficult to predict how stable/robust the new
meaning generated in these sense-making episodes will be. For example, even though
Oliver reasoned about both the infinite well and the bead-on-a-string, and came to
reject the infinite well representation for the bead-on-a-string. Unlike Quinn, Oliver
did not explicitly reflect on the differences of the two entities.
This concern leads us back to the arguments of Bransford, Schwartz and
Sears, (Bransford, Schwartz and Sears; 2005). In their double transfer studies, they
are arguing for early invention as a way to better prepare students to learn from
common resources. In their 2004 work, they show that students in the invent-a-
method condition and the tell-and-practice condition performed (Schwartz & Martin,
2004). However, assessment of the two groups after a common learning resource11
showed that the invent-a-method group was better positioned to learn from the
11 The common resource came in the form of a worked example on an exam.
148
resource. This suggests that early invention of toy models may not reveal immediate,
obvious learning gains. However, those opportunities may better allow students to
make sense of lectures and homework problems later on. And so, I implore
instructors to allow students opportunities to go past ‘fundamentals’, where they are
able to creatively leverage what they’ve learned to make sense of systems that are of
real interest to the,
149
References
Ambrose, B. S. (1999). Investigation of student understanding of the wave-like
properties of light and matter (Doctoral dissertation).
Baily, C., & Finkelstein, N. D. (2009). Development of quantum perspectives in
modern physics. Physical Review Special Topics-Physics Education
Research, 5(1), 010106.
Baily, C., & Finkelstein, N. D. (2010). Teaching and understanding of quantum
interpretations in modern physics courses. Physical Review Special Topics-
Physics Education Research, 6(1), 010101.
Bao, L. (1999). Dynamics of student modeling: A theory, algorithms, and application
to quantum mechanics.
Bao, L., & Redish, E. F. (2002). Understanding probabilistic interpretations of
physical systems: A prerequisite to learning quantum physics. American
Journal of Physics, 70(3), 210-217.
Bolden, G. B. (2009). Implementing incipient actions: The discourse marker ‘so’ in
English conversation. Journal of Pragmatics, 41(5), 974-998.
Bransford, J. D., & Schwartz, D. L. (1999). Chapter 3: Rethinking transfer: A simple
proposal with multiple implications. Review of research in education, 24(1),
61-100.
Brizuela, B. M. (2004). Mathematical development in young children: Exploring
notations. Teachers College Press.
150
Brookes, D. T., & Etkina, E. (2007). Using conceptual metaphor and functional
grammar to explore how language used in physics affects student learning.
Physical Review Special Topics-Physics Education Research, 3(1), 010105.
Carraher, D., Schliemann, A. D., & Brizuela, B. M. (2000, October). Early algebra,
early arithmetic: Treating operations as functions. In Presentado en the
Twenty–second annual meeting of the North American Chapter of the
International Group for the Psychology of Mathematics Education, Tucson,
Arizona.
Carraher, D., Schliemann, A., & Brizuela, B. (1999). Bringing out the algebraic
character of arithmetic. In AERA Meeting (40 pp.).
Cataloglu, E., & Robinett, R. W. (2002). Testing the development of student
conceptual and visualization understanding in quantum mechanics through the
undergraduate career. American Journal of Physics, 70(3), 238-251.
Daoud, M., & Hussin, V. (2002). General sets of coherent states and the Jaynes–
Cummings model. Journal of Physics A: Mathematical and General, 35(34),
7381.
Derry, S. J., Pea, R. D., Barron, B., Engle, R. A., Erickson, F., Goldman, R.,& Sherin,
B. L. (2010). Conducting video research in the learning sciences: Guidance on
selection, analysis, technology, and ethics. The Journal of the Learning
Sciences, 19(1), 3-53.
diSessa, A. A. (1991). Epistemological micromodels: The case of coordination and
quantities. Psychologie génétique et sciences cognitives, 169-194.
151
Domert, D., Linder, C., & Ingerman, Å. (2004). Probability as a conceptual hurdle to
understanding one-dimensional quantum scattering and tunneling. European
Journal of Physics, 26(1), 47.
Dósa, K., & Russ, R. (2016). Beyond Correctness: Using Qualitative Methods to
Uncover Nuances of Student Learning in Undergraduate STEM Education.
Journal of College Science Teaching, 46(2), 70.
Einstein, A. (1936). Physics and reality. Journal of the Franklin Institute, 221(3), 349-
382.
Emigh, P. J., Passante, G., & Shaffer, P. S. (2015). Student understanding of time
dependence in quantum mechanics. Physical Review Special Topics-Physics
Education Research, 11(2), 020112.
Gibson, J. J., & Gibson, E. J. (1955). Perceptual Learning: Differentiation or
enrichment. Psychological Review, 62, 32-51.
Gire, E., & Price, E. (2015). Structural features of algebraic quantum notations.
Physical Review Special Topics-Physics Education Research, 11(2), 020109.
Goodwin, C. (1994). Professional vision. American anthropologist, 96(3), 606-633.
Greeno, J. G., & Hall, R. P. (1997). Practicing representation. Phi Delta Kappan,
78(5), 361.
Griffiths, D. J. (2016). Introduction to quantum mechanics. Cambridge University
Press.
Hammer, D. (2000). Student resources for learning introductory physics. American
Journal of Physics, 68(S1), S52-S59.
152
Hatano, G. Inagaki, K. (1986). Two courses of expertise. Child development and
education in Japan. NY: WH Freeman.
Hatano, G., & Oura, Y. (2003). Commentary: Reconceptualizing school learning
using insight from expertise research. Educational researcher, 32(8), 26-29.
Hutchins, E. (1995). Cognition in the Wild. MIT press.
Hutchins, E., & Klausen, T. (2000). Distributed Cognition in an Airline Cockpit.
Cognition and Communication at Work, 15–34.
https://doi.org/http://dx.doi.org/10.1017/CBO9781139174077.002
Intuitive Quantum Physics, (http://umaine.edu/per/projects/iqp/)
Johnston, I. D., Crawford, K., & Fletcher, P. R. (1998). Student difficulties in
learning quantum mechanics. International Journal of Science Education,
20(4), 427-446.
Jordan, B., & Henderson, A. (1995). Interaction analysis: Foundations and practice.
The journal of the learning sciences, 4(1), 39-103.
Kalkanis, G., Hadzidaki, P., & Stavrou, D. (2003). An instructional model for a
radical conceptual change towards quantum mechanics concepts. Science
education, 87(2), 257-280.
Ke, J. L., Monk, M., & Duschl, R. (2005). Learning Introductory Quantum Physics:
Sensori‐motor experiences and mental models. International Journal of
Science Education, 27(13), 1571-1594.
Liboff, R. L. (2003). Introductory quantum mechanics. Pearson Education India.
153
Lin, S. Y., & Singh, C. (2009). Categorization of quantum mechanics problems by
professors and students. European Journal of Physics, 31(1), 57.
Mannila, K., Koponen, I. T., & Niskanen, J. A. (2001). Building a picture of students'
conceptions of wave-and particle-like properties of quantum entities.
European journal of physics, 23(1), 45.
Marshman, E., & Singh, C. (2015). Framework for understanding the patterns of
student difficulties in quantum mechanics. Physical Review Special Topics-
Physics Education Research, 11(2), 020119.
McKagan, S. B., & Wieman, C. E. (2006, February). Exploring student understanding
of energy through the quantum mechanics conceptual survey. In AIP
conference proceedings(Vol. 818, No. 1, pp. 65-68). AIP.
McKagan, S. B., Perkins, K. K., & Wieman, C. E. (2008). Deeper look at student
learning of quantum mechanics: The case of tunneling. Physical Review
Special Topics-Physics Education Research, 4(2), 020103.
McKagan, S. B., Perkins, K. K., & Wieman, C. E. (2008). Why we should teach the
Bohr model and how to teach it effectively. Physical Review Special Topics-
Physics Education Research, 4(1), 010103.
McKagan, S. B., Perkins, K. K., & Wieman, C. E. (2010). Design and validation of
the quantum mechanics conceptual survey. Physical Review Special Topics-
Physics Education Research, 6(2), 020121.
Morgan, J. T., Wittmann, M. C., & Thompson, J. R. (2003). Student understanding of
tunneling in quantum mechanics: examining survey results.
154
Müller, R., & Wiesner, H. (2002). Teaching quantum mechanics on an introductory
level. American Journal of physics, 70(3), 200-209.
Nanda, K. K., Kruis, F. E., Fissan, H., & Behera, S. N. (2004). Effective mass
approximation for two extreme semiconductors: Band gap of PbS and CuBr
nanoparticles. Journal of applied physics, 95(9), 5035-5041.
Olsen, R. V. (2002). Introducing quantum mechanics in the upper secondary school: a
study in Norway. International Journal of Science Education, 24(6), 565-574.
Ono, Y., Zimmerman, N. M., Yamazaki, K., & Takahashi, Y. (2003). Turnstile
Operation Using a Silicon Dual-Gate Single-Electron Transistor. Japanese
Journal of Applied Physics, Part 2: Letters, 42(10 A).
Parnafes, O. (2007). What does “fast” mean? Understanding the physical world
through computational representations. The Journal of the Learning Sciences,
16(3), 415-450.
Passante, G., Emigh, P. J., & Shaffer, P. S. (2015). Examining student ideas about
energy measurements on quantum states across undergraduate and graduate
levels. Physical Review Special Topics-Physics Education Research, 11(2),
020111.
Petri, J., & Niedderer*, H. (1998). A learning pathway in high‐school level quantum
atomic physics. International Journal of Science Education, 20(9), 1075-1088.
QuILT, (https://www.physport.org/examples/quilts/index.cfm).
155
Rebello, N. S. (2009, November). Can we assess efficiency and innovation in
transfer?. In AIP Conference Proceedings(Vol. 1179, No. 1, pp. 241-244).
AIP.
Rosebery, A. S., Ogonowski, M., DiSchino, M., & Warren, B. (2010). “The coat traps
all your body heat”: Heterogeneity as fundamental to learning. The Journal of
the Learning Sciences, 19(3), 322-357.
Sadaghiani, H. R. (2005). Conceptual and mathematical barriers to students learning
quantum mechanics (Doctoral dissertation, The Ohio State University).
Sadaghiani, H. R., & Pollock, S. J. (2015). Quantum mechanics concept assessment:
Development and validation study. Physical Review Special Topics-Physics
Education Research, 11(1), 010110.
Sadaghiani, H., & Bao, L. (2006, February). Student difficulties in understanding
probability in quantum mechanics. In AIP Conference Proceedings (Vol. 818,
No. 1, pp. 61-64). AIP.
Schegloff, E. A. (2007). Sequence organization in interaction: Volume 1: A primer in
conversation analysis (Vol. 1). Cambridge University Press.
Schegloff, E. A., & Sacks, H. (1973). Opening up closings. Semiotica, 8(4), 289-327.
Schiffrin, D. (1988). Discourse markers (No. 5). Cambridge University Press.
Schliemann, A. D. (2013). Bringing out the algebraic character of arithmetic: From
children's ideas to classroom practice. Routledge.
Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and
instruction, 16(4), 475-5223.
156
Schwartz, D. L., & Martin, T. (2004). Inventing to prepare for future learning: The
hidden efficiency of encouraging original student production in statistics
instruction. Cognition and Instruction, 22(2), 129-184.
Schwartz, D. L., Bransford, J. D., & Sears, D. (2005). Efficiency and innovation in
transfer. Transfer of learning from a modern multidisciplinary perspective, 1-
51.
Serway, R. A., Moses, C. J., & Moyer, C. A. (2004). Modern physics. Cengage
Learning.
Sidnell, J. (2011). Conversation analysis: An introduction (Vol. 45). John Wiley &
Sons.
Siegler, R. S., & Crowley, K. (1991). The microgenetic method: A direct means for
studying cognitive development. American psychologist, 46(6), 606.
Singh, C. (2001). Student understanding of quantum mechanics. American Journal of
Physics, 69(8), 885-895.
Singh, C. (2008). Interactive learning tutorials on quantum mechanics. American
Journal of Physics, 76(4), 400-405.
Singh, C., & Marshman, E. (2015). Review of student difficulties in upper-level
quantum mechanics. Physical Review Special Topics-Physics Education
Research, 11(2), 020117.
Smith III, J. P., Disessa, A. A., & Roschelle, J. (1994). Misconceptions reconceived:
A constructivist analysis of knowledge in transition. The journal of the
learning sciences, 3(2), 115-163.
157
Steinberg, R. N., Oberem, G. E., & McDermott, L. C. (1996). Development of a
computer‐based tutorial on the photoelectric effect. American Journal of
Physics, 64(11), 1370-1379.
Steinberg, R., Wittmann, M. C., Bao, L., & Redish, E. F. (1999). The influence of
student understanding of classical physics when learning quantum mechanics.
Research on teaching and learning quantum mechanics, 41-44.
Stevens, R., & Hall, R. (1998). Disciplined perception: Learning to see in
technoscience. Talking mathematics in school: Studies of teaching and
learning, 107-149.
Tutorials in Physics: Quantum Mechanics,
(https://depts.washington.edu/uwpeg/tutorials-QM)
Wittmann, M. C., Morgan, J. T., & Bao, L. (2005). Addressing student models of
energy loss in quantum tunneling. European Journal of Physics, 26(6), 939.
Wittmann, M. C., Steinberg, R. N., & Redish, E. F. (2002). Investigating student
understanding of quantum physics: Spontaneous models of conductivity.
American Journal of Physics, 70(3), 218-226.
Zhu, G., & Singh, C. (2012). Improving students’ understanding of quantum
measurement. I. Investigation of difficulties. Physical Review Special Topics-
Physics Education Research, 8(1), 010117.
Zhu, G., & Singh, C. (2012). Improving students’ understanding of quantum
measurement. II. Development of research-based learning tools. Physical
Review Special Topics-Physics Education Research, 8(1), 010118.
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Chapter 4: Designing Disruptions to Wavefunction
Infrastructure for Flexible Representation Use
Introduction
Developing proficiency with representations such as graphs, equations,
schematics, diagrams, and free-body diagrams, is central to learning and problem-
solving in physics (Kohl, Rosengrant, & Finkelstein, 2007; Ainsworth, 1999; Heller,
Hollabaugh, 1992; Fredlund, Airey, & Linder, 2012). However, representation
reproduction and use is also often challenging for students (McDermott, 1987;
McKagan & Wieman, 2006). My goal is not to solve these difficult instructional
issues, but rather, to problematize what counts as proficiency in using graphical
representations. Typically, research on instructional interventions deems students
proficient with a representation if students can demonstrate the skills of correctly
generating the representation and extracting information (reading out) from the
representation. Recent research, however, has started to build a more nuanced story of
students’ use of representations during sense-making, going beyond a “skills
development” narrative. For example, Heckler (2010) shows that requiring students to
draw diagrams can lead to a conceptual disconnect between the representation and the
student’s sense-making. Students who construct the diagram unprompted tend to do
more effective at integrating the diagram with conceptual reasoning. Others, for
example, Gire and Price (2015) and Parnafes (2007), have shown that different
physical features within representations may serve different roles in student
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reasoning. Gire and Price suggest that different algebraic forms may provide different
types of support for computation. Similarly, Parnafes shows that attention to
particular figural features of a situation or representation is important for coordinating
conceptual change. The picture that emerges from these studies is that representation
use by physics learners is contextual and tied closely to their conceptual reasoning.
In this chapter, I extend the thread of research on the dynamics of students’
sense-making and inscription use. I draw on Greeno & Hall’s (1997) framework for
describing representational practices students should engage in, which I term flexible
representation use for brevity. Drawing on existing literature helps direct my research
into fine-grained characterization flexible representation use in the context of
introductory quantum mechanics (e.g. Greeno & Hall, 1997; Lehrer, Schauble,
Carpenter, & Penner, 2000.) Taken together, flexible representation has the following
characteristics: 1) representations play an integral role in communication and
problem-solving, 2) construction, invention, or adaptation of representations to serve
local needs, 3) representations can co-evolve, or evolve reflexively, with evolving
conceptual understanding, and 4) the coordination of different representations.
To study flexible representation use, I draw on the notion of designing
disruptions to representational infrastructure to help guide the design of interview
protocols (Hall, Stevens, & Torralba, 2002). In those protocols, I “tweaked” the
traditional particle in a box scenario in introductory quantum physics to see how
students would use representations when sense-making in similar situations
(McKagan & Wieman, 2006). I do a microanalysis (Siegler & Crowley, 1991) of two
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episodes from my interview with a student, pseudonym Chad. Chad, I argue, creates a
rich system of representations to serve various coordinating, communicating, and
reasoning purposes. Chad generates and uses the representations reflexively, in the
sense that his conceptual thinking and his representations change in ways that
mutually affect each other. My goal with this work is to begin to describe and
characterize flexible representation use of inscriptions, which in this case, challenges
traditional notions of proficiency in using representations. These preliminary results
will inform future work on analyzing and supporting student sense-making, as well as
instructional design of representation tasks.
Greeno and Hall’s invocation to teachers
Greeno and Hall (1997) encourage instructors to provide students with a rich
variety of representational experiences because through participation, students learn
the predominant ways of knowing and learning of that setting (Greeno & Hall, 1997).
A broader variety of experiences can lead to a broader understanding of
representation use by students. Looking across studies of professionals and students
working with representations provides a sense of the breadth of experiences or
representational practices that people and communities can engage in.
Unfortunately, some classrooms fail to provide students with such a rich
variety of experiences and treat representation use very rigidly. Representations are
treated as end-goal in themselves. Here, students may be asked to reproduce or read-
out from standard forms introduced in instruction. Even though these practices may
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have good intentions behind them, they position representation-use as mainly a
performance to be evaluated on.
In other cases, instruction can provide opportunities for students to engage in
flexible representation use, where students participate in a variety of practices of
representation, including opportunities to invent, construct, interpret, and discussions
of conventions of use. Participation in a wider range of practices positions
representations as integral parts of sense-making and communication. Often a
nonstandard form can better serve the needs that emerge during communication and
sensemaking. This is because different representations may provide different supports
for reasoning (Greeno & Hall, 1997; Ainsworth, 1999; Gire & Price 2015).
In this realm of flexible representation use, inquiry with representations can
take on reflexive form:
“Solving a problem involves an interactive process in which students
construct representations based on partial understanding and then can use the
representations to improve their understanding, which leads to more refined
representations, and so on,” pg.365, Greeno & Hall, 1997.
What Greeno and Hall describe is a form of inquiry that is clearly non-linear. As
opposed to a process in which increasingly sophisticated ideas come to be inscribed,
the process of inscription is a driving mechanism in generating these increasingly
sophisticated ideas.
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With the rest of this chapter, I use this reflexive notion of inquiry as an
orienting characteristic of flexible representation use. It is a characteristic of student
sense-making that instructors may value because it supports the generation of a
deeper understanding of a situation. And so, in the following sections, I highlight
work others have done showing varying degrees of reflexiveness in student sense-
making with graphical or pictorial representations. In doing so, I try to highlight what
might be supporting or inhibiting this type of reflexive, connected sense-making.
Opportunities for Flexible Representation Use are needed in physics
In this section, I further motivate the need to study and create opportunities for
flexible use of representations. To do so, I make the following points:
1. Reflexiveness is a means pattern of sense-making that supports the
development of new meaning
2. More rigid use of diagrams and representations may lead to a disconnection
between representational and conceptual sensemaking
Current notions of reflexivity
Others have noted that conceptual and representational development often
takes on a reflexive characteristic. Kirsh (2010) provides a nice description of this
interactive process:
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“By ‘interactive’ I mean a back and forth process: a person alters the outside
word, the changed world alters the person, and the dynamic continues,” pg.
440.
Kirsh argues that by structuring the world in which we think, we can change
the “cognitive cost” required by my thinking processes. Such structuring makes the
thinking processes we perform simpler. Additionally, structuring the world through
external representations can also change what types of thinking processes are
possible. For example, the range of actions a thinker can perform on a representation
increases when the representation is externalized. The range of actions possible can
be extended further given that different representational forms can encode
information differently. And so, a thinker can “try out different internal and external
representational forms, the two forms can play off each other in an interactive
manner, leading to new insights,” (Kirsh, 2010).
A few others (Lehrer, Schauble, Carpenter, & Penner, 2000; Nemirovsky,
1994) have studied the reflexive and co-development of representations and
conceptual understanding. Both works point to the importance of attending to new
physical and representational features as important to, and emergent from, developing
representational and conceptual understanding.
Nemirovsky (1994) shows the joint conceptual and representational
development of the velocity sign by a high school student. In the study, the student is
developing ideas about the velocity sign through interaction with a computer program
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that allows her to move a cart on a track to plot distance and velocity of the cart. At
first, the student does not understand how to generate a negative velocity plot.
Attending to new physical and representational features, such as the starting and
stopping points, became crucial in her developing stories about the car’s negative-
velocity motion and her development of an understanding of how to generate such
representations (Nemirovsky, 1994).
Lehrer, Schauble, Carpenter, & Penner show the reflexive development of
conceptual and representational understanding in a third-grade science classroom
studying what aspects affect the growth of fast-growing plants. To do so, the students
developed a cascade of inscriptions (Latour, 1990) that helped shape their conceptual
growth. As their representations took on more dimensions, the students were able to
ask and answer different questions about the growth. For example, moving from
asking questions about how the height of the plant changes over time with one-
dimensional representations to asking questions about how the relationship between
height and width changes over time after developing two-dimensional
representations.
With this chapter, I hope to add to this small line of work showing co-
development of representational and conceptual understanding. In particular, my
work provides a complementary perspective by looking for reflexiveness on a finer-
grained scale. The development that takes place in the two works cited is much longer
than the span of a few minutes, which may be typical of a tutorial or homework
problem, where students are may be tasked with generating a single representation. I
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begin by reflecting on work that, unlike that described above, shows a deep
disconnect between conceptual and representational sense-making.
Rigid Use of Representations can lead to disconnected Sensemaking
It’s common practice in physics to ask or require students to draw a diagram
or representation when problem-solving. Many prescriptive problem-solving methods
start with having student draw a picture or representation (Heller, Keith, Anderson,
1992; Reif & Heller, 1982; Schoenfeld, 1985). These approaches are based on a
progressive translation from the problem situation to increasingly more mathematical
descriptions (Heller, Keith, Anderson, 1992). The first prompt often asks students to
draw a diagram or representation of the problem situation. However, some
researchers have found that prompting for representations may lead to sensemaking
that is disconnected from representation use (Heckler, 2010; Lehrer, Schauble,
Carpenter, Penner, 2000; Kuo, Hallinen, & Conlin, 2017). Prompting a representation
for assessment may fall under Greeno and Hall’s notion of “rigid representation use.”
Heckler studied a large group of students solving fairly standard introductory
mechanics problems. Heckler gave two groups of students identical problems on
identifying and modeling forces. One group was prompted to draw diagrams and the
other group was not. Analysis of the two groups’ problem-solving found that students
who were not prompted to draw diagrams were more likely to generate correct
solutions. Of the students in the two groups who drew diagrams, the prompted group
was more likely to generate solutions that were disconnected from their mathematical
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modeling. Additionally, the students who were not prompted to draw diagrams tended
to use more intuitive, less formal problem solving approaches. Heckler cites two
possible explanations: 1) novice students may be more effective at using more
informal, intuitive methods, 2) the act of prompting the diagram may cue
epistemological resources that treat drawing and mathematical modeling as separate
tasks (Hammer & Elby, 2003). In either case, prompting for the representation more
often led to disconnected representational and conceptual sense-making.
Two cases studies by Lehrer, Schauble, Carpenter, and Penner (2000) also
study the (dis)connectedness between drawing representations and conceptual or
mathematical sensemaking. However, they draw slightly different conclusions
explaining the source of this disconnectedness. In the two cases, elementary-aged
children worked on a unit exploring physical features that affect the speed at which an
object rolls down an inclined plane. In their inquiry, the students came upon the
problem of ensuring that all ramps were equally steep. The students had identified
three physical features of inclined planes that defined steepness: height, length, and
“pushed-outness.” One group was asked to invent a drawing that captured all three
features. The other group was given a representational form (triangle) and asked to
reason about the same issue (how the three features correspond to steepness). While
the first group was successful in the task, the latter group’s representations were
lacking in their ability to show how the three physical features affected the steepness
of the ramp. Lehrer et al. suggest that this group’s sketches were “less
representational, because we inadvertently began the discussion by providing the
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children with a solution to a problem that they had not yet accepted as problematic,”
(Lehrer, Schauble, Carpenter, and Penner, 2000).
In comparing to Heckler’s work, two important differences arise. First, Lehrer
et al. show the importance of giving students more choice in how they represent. For
example, the first group chose something that was only partially abstracted because it
included a crate propping up the ramp. However, this inclusion allowed the
manipulation of the height through changing the number of crates. From this, the
students could show how the height affected the steepness of the ramp. It also
suggests that there is some issue of timing in their sensemaking. The representation
should arise at the time and in a way to address something problematic in
sensemaking. It seems naive to assume that this need arises at the same time for all
students (i.e. in step 1 of a problem solving algorithm) or that it should be solved in
the same way for each student, such as with a formal free-body diagram (Heckler,
2010). Often, nonstandard representations may serve local needs better than standard
representations (Hall, 1996).
How do we approach creating opportunities for flexible representation use?
Designing Disruptions to Representational Infrastructure
Based on personal experiences with curricular materials and textbooks
(Griffiths, 2016; Serway, Moses, & Moyer, 2004; Liboff, 2003; Singh, 2008; Zhu &
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Singh, 2012, Tutorials in Physics12: Quantum Mechanics, QuILTs13, Intuitive
Quantum Physics14), most standard instructional materials require the use of
representations but don’t explicitly scaffold or focus on flexibility. This is
problematic given that research has shown that routine prompting of representations
can lead to disconnected sense-making (Heckler, 2010; Lehrer, Schauble, Carpenter,
& Penner, 2000; Kuo, Hallinen, & Conlin, 2017). Recent work on modeling
instruction (McPadden & Brewe, 2017) focuses on having students use a variety of
representations. However, none of these works focus on whether and how students
use representations flexibly.
In this section, we explore work from socio-cultural studies in mathematics
education to look for some answers to the problem of designing for flexible
representation use. In doing so, we encourage instructors to take a broad view of the
representations they ask students to produce in order to understand what opportunities
are available to students. Then we turn to literature on ‘disrupting’ representational
practices as a means of thinking about how to disrupt more routine representational
practices to create opportunities for students to use representations more flexibly.
12 https://depts.washington.edu/uwpeg/tutorials-QM 13 https://www.physport.org/examples/quilts/index.cfm 14 http://umaine.edu/per/projects/iqp/
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What is Representational Infrastructure?
In instruction, students may be asked to generate, sense-make around, or
communicate about different types of representation (i.e. wavefunctions in quantum
mechanics). Instructional histories, routines, and norms of practice around those types
of representations then support that generation, communication, or sense-making by
the students. This collection of histories, routines, and norms of practice are
considered a representational (Hall, Stevens, & Torralba, 2002) or informational
infrastructure (Star & Ruhleder, 1996).
Stable representational infrastructures have a certain scope in that they can,
and should, be used beyond a single instance. In doing so, the infrastructure should
support activity that follows the interests and norms of the community. The
infrastructure should support performing heavily standardized tasks efficiently, but
also have enough flexibility to be used in more customizable ways. For example,
representational infrastructure used for supporting the generation of wavefunctions
should be able to support students in quickly generating known wavefunctions for
familiar systems, but also in reasoning about the wavefunctions of unknown systems.
Use of a representational infrastructure can often rely on a high degree of
intersubjectivity between participants (Hutchins & Klausen, 2000). Those deeply
embedded within the community of practice may take this intersubjectivity for
granted. This can lead to situations with apparent intersubjectivity is reached, but
without actual deep understanding. Differences in understanding may simply go
unnoticed or unmarked. This raises the risk of misunderstanding of student actions by
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instructors. It may also mean that aspects of culturally sanctioned representations and
their infrastructures are not readily apparent to students.
From this work on representational infrastructure emerged a line of inquiry of
designing learning environments to disrupt more routine representational practices
towards enabling students to use representations in new, flexible ways. The collection
of work I review below suggests that designing disruptions to representational
infrastructure might guide instructors in creating opportunities for flexible
representation use.
Disruptions to Representational Infrastructure
Disruptions to representational infrastructure are bids for rejection,
replacement, or challenges (Hall, Stevens, Torralba, 2002) that require participants to
reorganize work practices to develop a new, or restructure existing, representational
infrastructure. With this section, I aim to make two main points:
1. Disruptions often lead to adaptation and invention, requiring some cognitive
and interactional work by participants to reorganize their work. The context of
the representational infrastructure and the specifics of the disruption help
shape the consequences of the disruption. The space may allow little
innovation and creativity or let innovation go unbounded, sometimes at the
expense of developing a stable infrastructure.
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2. Learning opportunities and access to representational practices are tied to the
representational infrastructure. Disrupting the representational infrastructure
can shift or redistribute opportunities for learning or maintain marginalized
access to the infrastructure.
Disruptions to Representational Infrastructure can serve to rebuke or better
align with community standards and norms
In looking at two case studies, Hall, Stevens, & Torralba (2002) focus on how
talk across disciplines helps shape disruptions to representational infrastructure and
how the participants then develop new representational infrastructures. The context of
the work being done in the two case studies puts some constraint or freedom on
possible courses of action. The two case studies examine an entomology group and an
architectural group.
In the “Bughouse,” they developed a routine of practices for gathering,
collecting, analyzing and representing data to make claims about the chemical
profiles of different termite species and colonies. The problem is that, while the group
is quite adept at looking across various representations to make these claims, the
process too lengthy in publication. And so the group invites a statistician to help
disrupt their routine, but also opaque, representational practices and develop ones that
are easier to present in publications. Disciplinary differences between the
entomologists and statisticians helps the BugHouse move away from using the
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awkward measures familiar to the entomologists and towards a more compact,
computational method.
The architecture group is tasked with remodeling a library that has been coded
as potentially unsafe through a representational infrastructure generated by the city.
Within the unsafe code, there are multiple paths forward in retrofitting the library so
that the building may be recoded as “safe.” The architecture group spends a lot time
of time discussing various pros and cons of different retrofitting options. All options
work to shift how the representational infrastructure would code the building. Instead
of working to fit the building to a new code, the group’s historian suggests a total
rejection of the coding scheme altogether. His suggestion is the representational
infrastructure is based on “arbitrary margins of safety,” and that a more realistic
infrastructure for coding buildings would classify the library as safe.
In comparing across the disruptions in the two groups, work within the given
contexts places different constraints on how the representational can be repurposed or
replaced. In both cases, the representational infrastructure embodies community
standards and norms. In the Bughouse group, the entomologists cannot reject the need
for concise, presentable data required of work in their professional community. Not
adhering to or working towards this disciplinary expectation would jeopardize their
very existence as members of that community. And so the representational
infrastructure for classifying termites is adapted to better serve those standards and
norms. In contrast, the architecture group is able to argue against community
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standards and norms. This is, in part, based on the historian’s experience seeing
similar “battles” fought in nearby cities.
Context of the representational infrastructure shapes opportunities for
adaptation and learning
In this section, I review other disruptions literature to discuss how learning
opportunities are tied to the representational infrastructure in routine and innovative
uses. Disrupting the representational infrastructure to develop more innovative
representational practice may help shift and redistribute learning opportunities. The
context of the disruption plays a strong role in determining how this redistribution
happens, if at all. In particular, a hierarchy of roles and differential access to aspects
of practice effect opportunities for both learning and taking agency towards the
innovation of the representational infrastructure.
Hutchins (1995) describes in great detail power loss in a Navy helicopter-
carrier coming into port. The power loss disrupts both the ability to slow the ship
down immediately and various technologies in the representational infrastructure for
coordinating the ship’s current position and in mapping what direction it is heading
in. The situation is dire. The team of navigators must quickly repurpose aspects of the
infrastructure (tools, technologies, forms of mutual monitoring) in order to find a
place for the ship to drop an emergency anchor. Loss of military and civilian life will
174
be unavoidable if the crew is unable to adapt the infrastructure to accomplish this end.
Three main changes in work characterize the crew’s adaptation to this disruption:
1. Routine practices for finding the ship’s orientation are ‘stretched’ in the
absence of quick feedback from downed technologies. The rudder angle is
turned more sharply than it normally would be in the case where the rudder
has electric power. With electric power, the rudder turns more quickly,
providing faster feedback to the navigators.
2. Practiced back-up mechanisms for accomplishing the task of navigation are
put in place. Instead of electric power turning the ship’s rudder, two crew
members turn large cranks to manually manipulate the rudder. There is no
time to find inventive solutions.
3. More experienced navigators take control of various posts. Under routine
practice, less experienced crew members usually take these positions, with the
experienced crew members providing feedback to help newer members learn.
And so, the life-threatening context of the distribution leads the crew members
to rely on well-established hierarchies of experience, backup technologies, and
adaptation of existing practices to safely anchor the ship. In doing so, they essentially
cut-off access to aspects of the navigational practice of various less-experienced crew
members. Learning is of no concern when lives are on the line.
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Like Hutchins, the case study on the Bughouse group by Hall, Lehrer, Lucas,
and Schauble (2004) shows how a hierarchy of roles tied to the representational
infrastructure shapes access to aspects of practice, both in routine and innovative uses
of the infrastructure. As described above, the representational infrastructure used by
the group coordinates various work practices and methods for analyzing and
representing data to describe the behavior of (or classify) various termite species and
colonies. Under routine use of the infrastructure, juniors coordinate field work while
the seniors are in charge of the research more broadly. The need to disrupt and
innovate the existing representational infrastructure arises from the need for
continued funding. Even in the process of innovation, the differential access in
determining the direction of the research is maintained. The seniors take charge in the
innovation and in doing so, intentionally create learning opportunities for the juniors
to understand the physical meaning behind the innovation. The seniors conceive of a
way to adapt existing work practices to make claims about when certain of species of
termites forage. The seniors help the juniors understand how these adaptations to their
work practices will help answer this question.
Like the example described by Hutchins, the group’s ‘livelihood’ is at stake,
but to a much lesser degree. The Bughouse group’s new line of inquiry needs to be
disciplined in the sense that they must, in a somewhat timely manner, devise of an
infrastructure that allows them to reliably address their new, fundable line of inquiry.
This is maybe, in part, why the seniors take the lead. However, the situation is not so
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dire that loss of life is imminent if a working infrastructure is not immediately
adapted. And so, seniors can take the time to make sure that the juniors are learning
through the innovation.
Unlike the above two examples, instructional design can help engineer more
distributed access to opportunities for learning and innovation (Ma, 2016; Hall,
Lehrer, Lucas, & Schauble, 2004). Instructional environments that require or support
innovative uses of infrastructure can shift the processes and content of student
learning. However, overabundance of opportunities for innovation can come at the
expense of opportunities for learning.
Ma worked with high school instructors to design Walking Scale Geometry; a
spatial disruption to typical geometry classrooms. Instead of constructing geometrical
shapes on a sheet of paper, students must invent tools and strategies for constructing
shapes that are the size of a classroom. The scale of the disrupted work requires
careful coordination between students and so learning becomes a joint activity. The
group of students collectively has agency towards developing new, innovative means
for solving previously mundane, individual problems. As the processes of learning
shift through development of these strategies, the content of what students learn also
shifts (Ma, 2016). For example, Walking Scale Geometry provides a particular
experiential context for developing an understanding of adjacent-side relationships
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and congruency because students are able to see and manipulate these constructs
through relative orientation with other students.
In Ma’s classroom, the need or desire to innovate representational practices
was not student-generated. The need was designed into the curriculum so that the
students would be forced to contend with it. In contrast, a second case study Hall et
al. (2004) documents a classroom where students have a different level of agency
towards innovation. The case study examines a 6th grade classroom seeking to
develop routine and innovative uses of a representational infrastructure of a system of
“pond jars,” used to answer questions about local pond ecology. The representational
infrastructure includes the “pond jars” themselves, processes of making
measurements, norms of representing finding to peers, etc.
The instructional set-up gave small groups of students agency over their own
inquiries. Every group was able to innovate with their own jars, or even repurpose the
entire class’s collection of jars to identify broader patterns during class-wide
“research meetings.” In some cases, unbounded innovation with the representational
infrastructure prevented disciplined inquiry. Students were unable to develop stable
enough infrastructures to answer their inquiry questions. In other cases, interaction
with the rest of the class and the instructional team helped students develop coherent,
non-confounding infrastructures to answer their inquiry questions.
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Takeaways from literature on Disruptions to Representational Infrastructure
1. Disruptions can lead to innovation and invention
2. The context of the disruption and differential access to aspects of practice can
effect opportunities for learning
3. Careful instructional design and interaction in instructional spaces can help
support innovation, invention, and disciplined inquiry
Designing for disruptions to representational infrastructure for generating
Wavefunctions
Disruptions to encourage flexibility in wavefunction representational
infrastructure
Wavefunctions are important in studying quantum systems because
wavefunctions contain all knowable information about the system. In this work, my
focus is the on the representational infrastructure associated with developing
wavefunctions for elementary particles. I am not interested in disruptions that
completely replace or reject representational infrastructure. Instead, I am interested in
disruptions that encourage more of a ‘stretching’ of the representational infrastructure
to include new contexts.
In the rest of this chapter, I will briefly describe the design of ‘disruptive’
interview tasks. After discussing analytical methods for studying student thinking on
these tasks, I’ll recount two episodes with Chad. At the time of these interviews,
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Chad was a senior physics major. Following these episodes with Chad, I will
highlight the characteristics and flexible representation use in the data. Though I do
mention instructional implications in this chapter, more can be found in the
concluding chapter of this dissertation (Chapter 5).
Data Context
My data were selected from a collection of 14 hour-long interviews with a
mix of physics and engineering students. Most students were of junior or senior
standing. In the interviews, the students were given tutorial-style problem sets and
were asked to think aloud with the interviewer as they went through the problem sets.
If there were moments of silence interviewer would check in. Questions directed to
students were meant to make the students' thinking more explicit.
The interviews were taken in two sets, with similar interview protocols for
each set. On the first set of interviews, the protocol included a problem on an infinite
well with a slanted problem, which I will refer to as the “slanted well” problem
(described in more detail below) (McKagan et al., 2010, Cataloglu & Robinett, 2002).
When viewing these interviews, I noticed a relatively consistent pattern of students
finding ways to adapt the representation of the infinite square well to reason about the
shape of the slanted well wavefunction. I then designed an interview protocol
explicitly around creating opportunities for this type of sense-making. In the design of
the second interview protocol, students first encountered problems on the particle in a
box; these included drawing energy eigenstates, describing the energies, and thinking
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about the speed of the particle. The next problem is on a classical analog of the
particle in a box, the “classical well” problem (described in detail below). Following
problems on the protocol include explaining quantum systems to peers, the slanted
well problem, and several problems on tunneling. Evidence of students adapting
representations occurred on the classical and slanted well problems. However, not all
students were able to get to these questions in the protocol. Across both interview
sets, I had a total of 17 clips or instances of students reasoning about the slanted and
classical well problems. The interview protocols can be found in the appendices.
Classical well problem
Problem statement: Suppose you had a classical particle in a physical
situation analogous to the quantum particle in the box. Consider a bead on a
string, and the string is knotted at x=0 and x=L so that the bead is confined
between 0 to L, and can move smoothly and freely between these bounds. The
bead has some energy E, and can bounce elastically at the knotted ends.
Sketch the wavefunction of this classical particle.
Opportunities for flexible representation use: Traditionally, wavefunctions
are used to describe the quantum state of a system. The wavefunction can be
used to describe possible outcomes of different types of measurements, such a
position of the particle or energy of the particle. In this problem, students are
asked to apply this quantum formalism to a fundamentally different type of
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system, a bead-on-a-string. Because the physical behavior are so different
between quantum and classical entities, students will likely have to adapt their
quantum practices to this classical context.
Solution: A flat wavefunction showing equal probability at every location.
Slanted well problem
Problem statement: Consider a quantum system with V(x) = ∞ for x=(-∞, 0)
and (L, ∞), and V(x)= Ax for x=(0, L). Sketch the wavefunction for the first
allowed state, or ground state, of the particle.
Opportunities for flexible representation use: I anticipated this problem
may encourage students to adapt because finding the wavefunction directly
from the Schrodinger Equation is not easy. Doing so would require students to
recognize the type of differential equation emerging from the Schrodinger
equation, which I think is not widely familiar to undergraduate students. Also,
the slanted well problem is not a heavily routinized example (like the infinite
well). For these reasons, I suspected that the slanted well problem would
encourage to adapt what they’ve learned.
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Solution: The canonical solution for the ground state is a first-order Airy
function.
Analytical Flow
Identifying relevant cases
I aim to model the reflexive development of conceptual and representational
meaning in student sense-making. My orienting research questions are then:
How does the student’s sensemaking shape the inscriptional space?
How does the inscriptional space shape the student’s sensemaking?
And so, my first pass at the data was to find relevant cases to further study flexible
representation use, using reflexiveness a broad-scale structure to look for. In doing so,
I looked for episodes of sense-making where there appeared to be several turns of
back-and-forth between conceptual reasoning and inscriptional development. I settled
on two episodes with Chad because of his talkativeness in these episodes. My hope is
that with these two episodes, I can begin to highlight key aspects of flexible
representation use.
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Analytical Methods15
Orientation: Looking for coordination among a system composed of people and
media
I tend direct my analytical focus ‘from the outside, in’. I take a systems-level
perspective first, then model how different aspects of the systems are coordinated to
generate the coherences exhibited by the system. i.e. What shared meaning is being
generated? How does interaction between the participants and their
drawings/representations help generate that meaning?
I take a systems-level perspective as primary because I see the student’s social
and material environment as playing a strong role in supporting student thinking. So
much so that the cognitive work required of the student can be significantly different
than the “cognitive work” accomplished by the entire system. An external
representation is not only a form of external memory, it can be a computational
medium (Hutchins 1995, Kirsh 2010). Take the example of the naval nomograph (fig.
4.1):
15 The analytical methods are adapted from the previous chapter: Learning with and about toy models
in QM
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FIG. 4.1 Naval nomograph
The nomograph consists of logarithmic line numbers for coordinating any two
of a ship’s speed, distance traveled over an amount of time, and the amount of time to
find the third quantity. For example, a navigator may take a straight-edge to the
nomograph, lining up the edge with a known travel time and speed to find the
distance the ship has traveled in that time. At a systems-level perspective, some
“cognitive” work has been done to accomplish this computation of speed. However,
the cognitive work that the navigator does is not computational, but more so focused
on the manipulation of the tool and perceptual pattern matching (Hutchins & Klausen,
2000; Rumelhart, Smolensky, McClelland, & Hinton, 1986).
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And so I start with a systems-level perspective to understand how integral
parts of the system -- people and drawings they create -- are working together in the
process of meaning-making. I only provide conjectures about what’s going on in
heads of participants when obvious mental transformation is functional towards the
meaning being made. This avoids the issue of overattributing cognitive process to
reasoning (Hutchins, 1995), either due to failing to confront the fact that cognition is
embedded in socio-material settings or that our own cultural embeddedness can affect
the meaning we (as instructors or researchers) attach to students’ reasoning (Star and
Ruhleder, 1996). The most appropriate unit of analysis is then studying talk-in-
interaction16 to understand what shared meaning the participants are developing
through coordination with each other and material structure such as drawings
(Hutchins, 1995).
The analytical question becomes: how do I study talk-in-interaction to model
meaning being generated?
1) look to the contents of interaction and speech to determine conceptual or
representational constructs
2) look at the structure and organization of the conversation provides a means for
understanding the functional meaning of those conceptual constructs
16 This is contrast to taking internal cognitive processes as a unit of analysis.
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Conversations can be organized and structured at various grain-sizes. For example,
single words or sounds may help meaningfully connect different clauses in a single
utterance. Or, turns in a conversation may be oriented towards the broader
conversational goal. Below, I provide a list of conversational structures/means for
organization that I attend to in my analysis.
17Progression; on a large scale, there is a sense that the conversation should
move towards accomplishing the mutually determined purpose
Turn-taking; individual turns at conversation
Repair; attempts to alleviate conversational trouble or breakdowns in mutual
understanding
Turn construction; conversational turns are structurally comprised of turn-
construction units, which may be single words, clauses, questions, etc.
Adjacency pairs (Sidnell, 2010; Sacks and Schegloff, 1973); distributed
conversational sequence of two utterances, where the first-pair part mutually
constrains second-pair part
Discourse markers (Schiffrin, 1988; Bolden, 2009); words such as “so”, “oh”,
“well”, “okay”
Example Analysis
17 This list is adapted from the previous chapters.
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Here, I provide a short snippet of data to show how I analyze the content and
structure of the interaction to understand meaning being generated. In the clip, “Paul”
is reasoning about the slanted well problem. The left column contains the transcript,
the middle column contains the inscription he is drawing or referencing in the
associated transcript. The third column highlights things I am noticing in the data. In
particular, I highlight; 1) how material structure becomes attached with conceptual
meaning and 2) the reflexive generation of meaning.
A more detailed example of analysis can be found in Chapter 3.
Transcript18 Inscriptions Noticing
Paul: Ok. So it would be more
likely to be found over here
((gestures to left side)), so that
means you would get
something kind of like....
Gesture to part of the inscription ‘attached’
conceptual meaning to the left side.
“So that means” shows some causal
connection between the area of least
likelihood and the wavefunction. Context
18 Transcript conventions can be found in the appendices.
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Something kinda like... tha:t?
((draws wavefunction))
leads us to infer what he is drawing is the
wavefunction.
Interviewer: So, can you
explain the different parts that
you've drawn?
Interviewer positions the wavefunction Paul
has drawn has having different “parts.”
Paul: So, it has to be zero on
the edge ((points to left edge)).
And then, I guess, and then it
would have to asymptotically
approach zero over here
because the potential gets
higher.
Right here ((points to peak)),
it's kind of we:ird.
Paul takes up the piece-wise treatment of the
wavefunction in his explanation, explaining
three different parts.
New conceptual and representational
meaning is emerging in his explanation;
asymptotic behavior on the right side is due
to the raising potential.
Paul highlights the peak as “weird,” and will
go on to sense-make about it in the rest of
the episode (not shown). The representation
he has drawn is shaping his conceptual
sense-making in that it necessarily contains a
feature that he does not understand. And so
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the representation has introduced a new
feature that he must sense-make about.
Table 4.1 Example analysis of episode in interview with Paul
Now, I move to discuss my focal episodes from the interview with Chad. In
particular, I aim to demonstrate the reflexiveness in Chad’s sense-making on the two
different prompts. In showing the ways in which Chad’s sense-making shapes, and is
shaped by, his representation(s) I highlight important features/processes in his sense-
making that support reflexiveness in his sense-making.
The first episode analyzes Chad’s sense-making on the classical well problem.
The second episode analyzes Chad’s sense-making on the slanted well problem. Each
episode is split into several subsection. The subsections try to show at least one
feedback loop in reflexiveness. Each loop shows how Chad’s understanding becomes
inscribed and how that inscription, or process of inscribing, comes to play an integral
in Chad uncovering new meaning about the system.
Chad on the Classical Well Problem
Chad’s modeling of the bead-on-a-string focuses on the knotted ends
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After reading the prompt, Chad moves from sense-making about the knots
constraining the bead to reasoning about what these constraints physically mean for
the bead’s wavefunction.
Chad: Let's start drawing.
((Draws shading in probability representation, box
lines, string))
So this is L.
((labels L in probability representation))
We have, knots on the, edges, right?
((traces prompt))
Interviewer: Mhm.
Chad: Does it say that? The thing is knotted, at, so yeah, right.
((traces lines of prompt and draws
knots))
So it can bounce at the knots, and it has some energy E. Bounce elastically, so
it keeps all of its energy. So that means.... it's bouncing, ela:stically, but yeah
it has to be accelerated, so it does slow down still. Um, yeah. So, that means,
the wavefunction is the thing that squared would be the probability.... It has to
look like this.
((draws wavefunction above axis
in probability representation)).
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FIG. 4.2 Chad’s Probability Representation
Chad moves from referencing the problem set-up to sensemaking about what
the setup means physically for the bead. In doing, Chad focused on marking the knots
at the edges and what the bounce means physically. He reasons that the elastic bounce
means that the bead keeps all of its energy. The bead also “has to be accelerated, so it
does slow down still.” For Chad, this implies that the wavefunction resembles what is
shown in Fig. 4.2.
It’s not yet clear how the bead’s acceleration and speed imply a wavefunction
or a probability density for Chad. Or how a constant energy may help shape the
wavefunction. In any case, it’s apparent from Chad’s drawing and reasoning that the
knots on the string are so far an important focus in his sensemaking. It may also seem
strange that Chad clearly marks the knots on a plot for the wavefunction. However,
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this is reminiscent of the cultural norm of plotting a quantum particle’s potential in
same space as its wavefunction. Work in that type of blended space may allow Chad
to visualize how the ways in which the bead is confined or constrained gives rise to
physical/wavefunction behavior.
Construction of the Position Representation of the Bead-on-a-string initiates a
change in the Wavefunction Representation
Following his drawing of the wavefunction, the interviewer asks for an
explanation of where the wavefunction came from. This sets of a new chain of
reasoning. Chad introduces a representation of the bead’s position over time to
explain his wavefunction to the interviewer. In doing so, he comes to realize that his
original wavefunction is not quite right. New representational (and associated
conceptual meaning) are then generated through explaining his position
representation.
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FIG. 4.3 Chad’s Position Representation
Interviewer: So, can you explain where that comes from?
Chad: Well, um, so the wavefunction, always has to be such that psi star psi is
the probability distribution.
((writes on
paper psi star psi equal box probability))
Interviewer: Ok.
Chad: Right?
Interviewer: Mhm.
Chad: So, if you start with the probability distribution, that, it's going to be
spending the majority of its time at the edges
((gestures to edges in probability representation))
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because, uh, it has to be accelerated at the edges.
So, it's essentially doing….. thi:s
((draws position representation))
if this is time, and this is x
((labels x and t above position representation)).
It's doing that. Which means that the majority of it is here,
((draws partitions in two position representation by drawing two loops
encompassing the edges)),
the minority of it is here.
((gestures to center of position representation))
So... it's not really, doesn't go to zero there
((erases center of the wavefunction))
Higher
((redraws center of wavefunction))
Chad: Alright, um. So if you have that, actually, technically, it could be also
be this
((Draws lower wavefunction in probability representation))
But it could never cross, because then it would go to zero.
((gestures over center of wavefunction))
So yeah. It has to be either this or this,
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((points to wavefunctions in probability
representation on “this” and “this”))
because psi star psi would be the probability distribution
((points to psi star psi equals probability))
which has to look like this.
((points to upper wavefunction in probability
representation))
So psi is plus or minus the square root of the probability
((write psi equals +- probability))
Interviewer: Ummm.
Chad: Almost.
((continues writing))
Interviewer: Oh, I see. Ok. So you're saying the probability looks, like, uh
((E points to upper psi in probability
representation))
Chad: ((Draws thin line of probability above wavefunction in probability
representation))
Interviewer: The probability...
((points to line Chad just drew))
ok. And then the square root, is like, either that one or that one
((pointers touch both wavefunctions in
1)).
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Chad: Yes, this would be psi plus this would be psi minus, and this one is psi
star psi.
((labels)) ((labels)) ((labels))
Chad: So, essentially deconstructing it from what the probability distribution
is
((gestures to probability distribution in
probability representation))
Interviewer: Ok, that makes sense.
Chad: Yeah.
Chad’s sensemaking is now attending to new areas of the bead’s motion, in
addition to the bounce. This attention helps generate new meaning (nonzero
probability) through close coordination of Chad’s two representations. Interestingly,
it seems that his focus on the bounce region naturally led him to point out the
complementary region (the center region), leading to new conceptual and
representational meaning being attached to the center.
Chad sets up his explanation for the wavefunction through the introduction of
the probability distribution. He partitions the position representation to highlight the
edge (“bounce”) region where the bead is accelerated and therefore spending the
“majority” it's time. Then, he moves his attention to the center region of his position
representation, highlighting the region where the bead spends the “minority” of its
time. With his attention on this center region, Chad comes to the sudden realization
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that something is amiss in the associated center region of his wavefunction
representation. The wavefunction should not go to zero in the center, it should be
somewhat higher. Chad moves to make the adjustment.
Turning back to his explanation initiates another change in the wavefunction
representation. Chad realizes that another, negative wavefunction is also possible and
so adds it to the representation. Chad’s explanation comes full circle in reiterating the
relationship between the probability distribution and wavefunction. In doing so, he
codes the upper wavefunction as the probability. This leads to some ambiguity as it
seems like Chad has now referred to the upper wavefunction in his representation as
both probability and the wavefunction. An implicit need for explanation arises as the
interviewer stumbles over the “probability,” leading to another representational
development. Chad adds another curve showing the probability distribution.
Chad’s Explanation of his Position Representation leads to a deeper
understanding of the bead’s energy and a refinement of the position
Representation
The interviewer again asks for an explanation. This time, of Chad’s position
representation. This again sets off a new chain of reasoning in which Chad sense-
makes around new areas and comes to a deeper understanding of the bead’s energy
and motion.
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Interviewer: Ok. Cool. Can I ask you, did, where this, did this come from
somewhere? Or is it.
Chad: Well, uhhh...
Interviewer: Besides your brain?
Chad: Yes, it did. Uh, because, it is, the:..
((pencil hovers over position representation))
O:h actually, it's not perfectly like that.
((pencil hovers over position representation))
Because this is the wavefunction for if, for if the relation is, the, the distance
away from it is
((writes diff equation x equals
negative double-dot x))
equal to the negative acceleration of it.
Interviewer: Oh, ok. That's your diff equation for the--?
Chad: Yeah, that's wave, essentially. Cus x dot dot plus x equals zero.
((writes diff equation x plus double-dot x
equals zero))
That would give you a wave. But this isn't exactly that because it's bouncing
off the edges
((points to
differential equation))
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But I assumed, fully elastic, it would have to have some sort of, uhhh some
sort of squishing element to it.
Interviewer: Uh huh, yeah.
Chad: To take in energy, because you can't just go
((gestures bead bouncing over
table, makes noise at the gestured bounce))
Interviewer: Yeah.
Chad: That
((holds hand in ball form in area where it hand bounced before in gesture))
Yeah.
Interviewer: So it kinda has to squish, and then reform.
Chad: Yeah. So it would, it has some energy E, which is normally in 1/2mv^2.
((Writes E
= 1/2mv^2)
But then it'll be so:me, let's make it E elastic kx involved.
((Adds +kx to energy equation))
Interviewer: Ok, cool.
Chad: Just for the edge
((darkens the knot on the right side in probability
representation)).
Because the knot, since it's not, not, we're talking about classical situations it’s
got some associated size, and bounceable-ness because it's elastic. So yeah, it
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has to, in, so this region would be flatter than how I drew it before. So, it'd be
like that.
((straightens wiggly lines inside of the vertical lines he drew in position
representation)).
Interviewer: Ok.
In looking at his position representation, poised for an explanation, Chad
again realizes that his representation is not quite right. Chad’s continued sensemaking
around the bounce at the edge provides a means of justifying why the position
representation does not accurately describe the bead-on-a-string through the
generation of a new understanding of the bead’s energy.
He first says that his (incorrect) position representation is described by his
written differential equation but cites the bounce at the edge as a reason to reject the
differential equation. In doing so, he implies his position representation needs further
development. That he cites the bounce, mentions acceleration in his description of the
differential equation, and then goes on to adjust the inside region of the position
representation may indicate that he’s realizing that the differential equation only
applies to the bounce region. We might infer this because Chad has mentioned the
acceleration at the edge, first somewhat implicitly and then more explicitly in
mentioning the “negative acceleration of it”.
In any case, he continues his algebraic modeling for the bead with the bounce
region coordinating sense-making. His mention of the bouncing at the edges as
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disqualifying the differential equation is followed by some consideration of how
energy is flowing at the bounce. Specifically, the energy takes kinetic form in the
center region with some additional elastic energy in the bounce regions. In doing so,
Chad writes a force instead of an energy. However, his intentions are clear and the
misstep seems inconsequential for his developing model of the situation.
Chad’s understanding of the bead’s energy is significantly developed from his
first modeling of the bead-on-a-string. In his first modeling, Chad simply reasoned
that bouncing elastically meant that it kept all of its energy. Here, Chad has deepened
his consideration of the energy, both in terms of what form the energy takes (elastic
and kinetic) and how those forms may differ in different regions of the bead’s motion,
through his sense-making which continuously coordinated by the bounce region.
From this, Chad flattens the lines in the center region of his position representation
(where the bead spends the “minority” of its time). This flattening follows
conceptually from his new modeling of kinetic energy in the center region where the
velocity is constant.
New meaning in the Position Representation leads to a new understanding of the
Probability and Wavefunction of the bead
Chad’s new understanding of the energy has a cascade of consequences
through his representational system. It leads to new conceptual and representational
understandings of the bead’s position, wavefunction, and probability.
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Chad: And then this bit is where
((Traces vertical line in position representation))
Oh yeah, so that's not quite right, is it?
((gestures to probability representation))
Interviewer: What made you say that?
Chad: Cus, the entire inside bit has equal probability.
((traces partitioning lines in position representation))
Because it spends the same amount of time here as it does here
((gestures to two points in
position representation, unclear which ones))
Interviewer: Mhm.
Chad: Because this is following the same velocity
((points to energy equation))
because, I'm saying the elastic collision only happens at the end bits
((traces area
outside of partitions in position representation))
And that's where it's accelerated, enough to make it go back to the same
velocity at this point
((marks points in position representation))
Interviewer: Ok.
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Chad: So that's, this vin and this is vout
((labels vin and vout in position representation)).
Interviewer: Ok. Sorry, so you're saying the velocity is the same along those
straight lines.
((pointer traces
lines in position representation))
And then, so can you relate that to probability?
Chad: Ye:s. It would make it, so that, it looks more like
((draws frame of revised
probability representation))
So it would be a flat line until the knot region, in which case it curls up,
((draws wavefunction in revised probability representation))
because where it's stopping is where it's spending the most time.
((dots left knot in probability representation))
So it's the highest probability at that point. And this point
((marks left peak of revised probability
representation))
((marks right peak in revised probability representation))
So, this is probability distribution.
((draws probability in revised probability representation))
So that makes, the psi plus and minus into.... flat li:ne
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((draws top wavefunction in revised probability
representation))
Just, essentially a smaller version of it, and a negative version of it.
((points to probability in revised probability representation))
((draws lower wavefunction in revised probability representation))
Interviewer: Ok, that makes sense. Cool.
Chad: Cool.
Interviewer: Yeah.
FIG. 4.4 Chad’s Revised Probability Representation
Chad’s explanation of his position representation quickly leads to new
conceptual and representational meaning in both his position and wavefunction
representation.
Chad is explaining the consequences of his new understanding of the bead’s
energy when, mid-explanation, when Chad’s attention is on the boundary line
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between the bounce and center regions, Chad realizes that his wavefunction is not
right. In the position representation, Chad explains the form of the probability in the
inside region by reference to two arbitrary two points, reasoning that the points have
equal probability because the velocity is constant in that region because the “elastic
collision” and acceleration only happens in the bounce region. From this, Chad draws
another set of wavefunctions.
Chad’s explanation has now come full circle. His original wavefunction was
reasoned through attention only to the bounce region; the acceleration at the bounce
meant the bead spent more time there, yielding a higher probability in that region.
This conceptual and representational understanding remains unchanged. However,
Chad has further fleshed out the conceptual and representational consequences of the
acceleration at the bounce through reasoning more carefully about the center region.
Here, the acceleration at the bounce puts the bead back to a constant velocity and
therefore constant probability in the center region.
Discussion of Chad on the Classical Problem
In this section, I illustrate key features of flexible representation use. In
particular I show the reflexiveness of between Chad’s developing conceptual
understanding and his representational system. Representations not only came to
reflect Chad’s sensemaking but also helped generate new lines of reasoning. In turn,
new lines of reasoning become embodied in his representational system. In this back
and forth, Chad’s sensemaking was continually coordinated by a focus on the bounce
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area. Highlighting and sensemaking around the bounce in his various representations
helped drive sensemaking in the regions complementary to the bounce, the center
region and the boundary.
Chad’s position representation served multiple roles. Sometimes the
representation was a tool for explaining and justifying his thinking. At other times,
the same representation became generative towards his sensemaking. I recount how
the close conceptual and perceptual coordination between the representations Chad
has drawn and his attention to particular areas in those representations help coordinate
his sensemaking, leading to new conceptual and representational meaning.
1. Construction of the Position Representation of the Bead-on-a-string initiates a
change in the Wavefunction Representation:
Explaining his wavefunction representation led to the generation of the
position representation and then the partitioning of the representation to highlight the
edge regions. Introducing a different representation of the bead helps Chad ‘see’ and
reason about regions of the bead’s motion that he had previously not considered.
Highlighting the complementary center region led to the realization that the
wavefunction should be non-zero in the center. Chad doesn’t fully flesh out why the
wavefunction can’t go to zero in the center. It seems that when he makes this
realization, it’s the first time in the episode that he is coordinating his visual
perception with the center region in the representation, along with the conceptual
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notion of where the bead is spending its time. It seems likely that in looking across
his plot of the bead’s position over time, he can easily read-out that every point is
being occupied by the bead at some point in time. Coordinating his notion of ‘more-
time yields a higher probability’ may allow him to quickly infer that the probability is
non-zero in the center.
2. Chad’s Explanation of his Position Representation leads to a deeper
understanding of the bead’s energy and a refinement of the Position Representation:
When the interviewer asks about the source of the position graph, Chad has to
reason more directly about the representation. In doing so, Chad describes his
deepening understanding of the bead’s energy to alleviate issues that became apparent
to him when he was poised to explain the position graph to the interviewer. The
consequences of this modeling lead him to the new conclusion that the velocity is
constant in the center of bead’s motion and therefore the position representation
should be flatter in the center region. This point shows how the using velocity
representation reflexively shaped Chad’s sense-making, leading to new conceptual
meaning about the bead’s motion.
3. New meaning in the Position Representation leads to a new understanding of
the Probability and Wavefunction of the bead:
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Chad’s new understanding of the center region in the position representation
becomes quickly generative towards developing new representational and conceptual
meaning in the wavefunction representation of the bead. The flatness of the position
representation implies a constant probability in the wavefunction representation. The
quick flow of consequences from new meaning coordinated in the position
representation to the wavefunction representation shows the close coupling of the
representational system Chad has developed to help coordinate his sensemaking.
Chad on the Slanted Well Problem
Chad recruits the Finite Well representation to reason about how potential walls
affect the wavefunction
After reading the prompt, Chad mentions that the slanted well was a problem
on one of his final exams, although it was not something they went over in class.
Although the situation is familiar to Chad, we see him constructing aspects of his
sense-making as the interaction unfolds. In his sense-making, Chad first draws on the
finite well representation to sense-make about how potential walls affect the
wavefunction.
Chad: ((Reads prompt))
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So another particle in a box. V=Ax oh. Oh. Ok, it's particle in a box.
((draws axes and box in slanted well
representation))....
Interviewer: Ummmm sooo I think it's maybe not quite--
Chad: Oh Ax. Sorry, I did not quite read that right. You're right.
((slanted well representation-- changes bottom
to slant, erases flat bottom))
It is. Alright, this was one of my final questions.
Interviewer: Was it really?
Chad: Yeah.
Interviewer: No wa:y.
Chad: Draw the states allowed.
Interviewer: Is this something you guys did in class?
Chad: No. It was only on the final. We talk about how the potential, uh walls
affect it
((traces
vertical wall on slanted well representation)).
And how it would be uhh...
((re-traces vertical wall on slanted well representation))
Yeah 'cus we talked, yeah if you talk about, uhh finite regions,
((draws well in finite well
representation))
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you have the wavefunction in here
((begins to draw wavefunction in finite well
representation)),
it doesn't go to zero here
((crosses boundary in finite well representation))
it goes to the points that it does, then exponentially decays in it
((draws decaying wavefunction in left, then right region))
Interviewer: I see.
Chad: And then... if they're tall enough you get
((adjusts potential walls in finite well representation to go down to V=0))
Tunneling!
((extends wavefunction in finite well representation))
Interviewer: Yay.
Chad: But yeah for this one it would just be, uhh it starts off like it and then it
decays
((ground state in finite well representation with matching speech,
draws n=2 with no comment))
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FIG. 4.5 Chad’s Slanted Well Representation
FIG. 4.6 Chad’s Finite Well Representation
At the beginning of his turn, Chad has a platform ready (the slanted well
representation) to hold his representation of the “states allowed.” Chad begins a chain
of causal reasoning in stating that the “potential walls affect it.” In doing so, he
highlights and re-highlights the vertical, right line in his slanted well representation,
bringing forth one aspect of the inscription as particularly relevant to the shared
visual field. The overlapping speech and tracing indicate that Chad is referring to the
vertical line as a “potential wall.” As the episode progresses, we see continued
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evidence that the regions and boundaries created by walls in his inscriptions structure
and coordinate his reasoning.
After an abandoned start, “and how it would be…,” Chad moves vertically
down his page and begins to draw the finite well representation, a standard
representation of a finite potential well. The finite well representation unfolds with
his speech; he notes that the wavefunction “exponentially decays within it.” The use
of the preposition “within” and his concurrent drawing of the wavefunctions in the
regions of high potential connects the idea of “decay” to regions bounded by potential
walls in the inscription.
In this short piece of speech, the physical structure of the potential walls feeds
into his sense-making in several ways. Looking at his utterance, once he begins to
expand on his causal reasoning of how the “potential walls affect it,” his speech is
naturally punctuated at the potential wall boundaries. The clauses of his speech either
refer to a region of the inscription (“uhh finite regions,” “you have the wavefunction
in here,” and “then exponentially decays in it”) or refer to the values at boundaries in
the inscription (“it doesn't go to zero here” and “it goes to the points that it does”).
His reasoning about the wavefunction is compartmentalized into reasoning about its
properties in regions and at boundaries. I would therefore infer that the representation
is playing some role in ‘bounding’ Chad’s reasoning, perhaps by supporting a causal
or descriptive account he is remembering, or partially generating on the fly, about
wavefunction behavior.
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Chad takes the outermost walls of the inscription and extends them further
down. When he adds to the wavefunction, he draws the piece of the wavefunction that
extends horizontally beyond the boundary walls that he had just previously extended.
In this brief turn, Chad recounts a causal story about the emergence of tunneling,
which can occur if these pieces of the wavefunction are “tall enough;” a condition
which does not have empirical conceptual underpinnings, but is more readily a
condition met by the physical form of the inscription. The first form of the inscription
(without extended walls and wavefunction), provides the setup conditions necessary
for the tunneling to occur. To be clear, the causal story of tunneling is told more
through the manipulation of the physical form of the inscription than readily apparent
conceptual ideas about the particle or probability. As before, the region-based
structure of the inscription structures Chad’s speech in his implicit treatment of the
wavefunction as piece-wise, through his coding of these pieces of the wavefunction
as “they”, where “they” end at the boundaries of potential walls.
Chad models the Slanted Well wavefunction as two competing influences: the
Particle in a Box wavefunction and the effect of the rising potential
The interviewer looks at the wavefunctions Chad and drawn and asks for an
explanation of the shapes. Chad draws two representations, the infinite-well
wavefunction and a slanted potential below. In doing so, Chad shows how the effect
of the rising potential can give rise to the slanted well wavefunction.
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FIG 4.7 Chad’s Infinite well representation above the slanted potential representation:
Chad draws diagonal line (slanted potential representation) directly below his infinite
well representation. Lettering below the slanted well representation occurred later in
the episode and so should be ignored.
Interviewer: So can you tell me like how you kno:w like, kinda of the shapes
of those guys?
((points to wavefunctions in slanted well representation))
Chad: So it's, you can kind of take it as perturbation on the particle in a box
((infinite well
representation))
So it's going to be essentially particle in a box
((draws n=1, 2 in infinite well representation))
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but then, uhh what's happening is as the potential increases
((potential slant representation))
it uhh, reduces the probability of being in that region.
Which means that if you still normalize it,
((traces n=2 in infinite well representation))
it would have to follow, it would have to follow the same energy, stepping,
where it's going by nodes added, but it will reduce the probability of this
region, linearly.
((circles right-hand side
of n=2 in potential slant representation))
Interviewer: Uhh, this region? Is that...
((E points to right-hand side of n=2 in slanted well representation))
Chad: Yeah, this is the potential.
((shades under slant in slanted well representation))
The new conceptual ideas emerging are that the problem is essentially a
perturbation on the usual infinite well, and the potential and probability are inversely
related. Again, it’s important to note that the relationship between potential and
probability is coordinated through reference to given “regions,” now referring to the
higher-potential vs. lower-potential parts of the interior of the “box.” The potential-
probability relationship he posited is also continuous with, and in some ways a
refinement of, his previous conclusion that the potential walls have an effect on the
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wavefunction. Chad’s verbal coordination of these two conceptual ideas leads to the
creation and manipulation of new inscriptions. The representational system grows to
include a common form of the infinite well representation with the potential slant
representation positioned directly below. His reasoning appears to structure and
organize the construction of the inscriptions through this vertical alignment: Chad can
more easily read-out the locations of areas of high potential, and therefore high effect
on probability. Chad’s speech initially sets-up the mediating relationship between two
conceptual ideas. However, it’s the strong visual coordination between inscriptions
the infinite well representation and the potential slant representation that provides the
platform for reasoning and showing that the right side of the wavefunction in the
infinite well representation will see reduced “probability of being in that region.” He
circles the region of interest in the infinite well representation, the interviewer then
confirms that he is implicitly also reasoning about the corresponding region in the
slanted well representation, to which he agrees. Ultimately, through this interaction,
Chad manipulates a standard, culturally-sanctioned form of infinite well
representation, through the mediating effect of the potential slant representation, in
order to draw conclusions about reduced probability in the right side of the ground
state in the slanted infinite well representation.
Modeling the Slanted Well wavefunction as two competing influences generates
a new conceptual and representational feature: the turning-point between the
two influences
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Chad continues to flesh out his modeling of the slanted well representation.
As he goes back and forth between conceptual and representational sense-making,
new representational features emerge as his sense-making develops.
Chad: But, uhh so it will follow essentially it
((traces part of ground state of the infinite well in slanted well
representation))
I think I made it too big for my waves to look right. But it will go into it
generally like that, ((points to n=slanted well representation in infinite well
representation)).
but it will also decay after it enters the region.
((traces remainder of wavefunction in slanted well representation,
leaving little line showing where two wavefunctions deviate))
The probability function looks a bit weird on this because it goes past zero and
comes out to it slowly.
((draws dotted line in
slanted well representation for n=2))
Chad: Looks like the....
((adds presumably psi squared for n=2 in slanted well
representation))
Interviewer: So is that psi squared for the... n=2?
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((points to n=2 in slanted well representation))
Chad: Yeah. That's psi star psi for... This is ...
((labels wavefunctions))
He traces the wavefunction as he attends to the deviation from the unperturbed
infinite well, leaving a little line that the normal ground state might follow, where the
slanted ground state begins to decay away.
In this concluding bit of speech, Chad finalizes his conceptual coordination of
the unperturbed infinite well and the mediating effect of the potential increasing. He
turns to the slanted infinite well, and shows how the state takes on characteristics of
the unperturbed infinite well and then begins to decay towards the right side of the
inscription. Although interpretations are possible whereby the various representations
merely express Chad’s thinking, we argue for a more reflexive relation between the
inscriptions and Chad’s thinking whereby the inscriptions influence and help shape
his thinking: the decaying wavefunctions inscribed in the forbidden regions of the
finite well representation, with forbidden corresponding visually to where the
potential is higher than the wavefunction, combined with the visual coordination of
the infinite well representation and the potential slant representation as discussed
above, contributes to Chad’s in-the-moment drawing/thinking about the lowest
wavefunction in the slanted well representation. The particular shape of the
wavefunction—unperturbed then decaying—as well as the conceptual insight that the
probability is lower where the potential is higher are constructed through the
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generation, manipulation, and reasoning with the inscriptions in the finite well
representation, the infinite well representation, and the potential slant representation.
Discussion of Chad on the Slanted Well Problem
In this brief episode, Chad’s sense-making and inscription use are deeply
entwined, occasionally unfolding together and other times feeding back into each
other reflexively. In some moments, his talk was structured in terms of the properties
of regions of the inscription. In some places, Chad’s judicious use of inscriptions may
provide additional structure to the conceptual ideas he brings up. For example, his
speech explicitly relays that the problem is like the infinite well, but that there is some
mediating effect of the potential increasing. However, the cognitive work to find the
region affected in the slanted well representation, where the wavefunction is “below”
the potential, is not accomplished in his speech, but through the coordination of other
inscriptions.
For Chad, different inscriptions do different work, and he generates new
inscriptions to serve specific, emergent purposes; even the culturally-sanctioned
representations (infinite and finite well representations) he draws do not simply serve
read-out purposes. Chad does work on these inscriptions to coordinate different
conceptual ideas, giving additional meaning to the non-standard form he constructs,
the slanted well representation. Not only do different inscriptions serve different
purposes for Chad, different parts of individual inscriptions play different roles. For
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example, the right side of the slanted well representation initiates the causal reasoning
about how the rising potential affects the probability, with the regions of
wavefunction decay in the finite well representation playing a role in that reasoning.
Comparing the two episodes
What’s similar? Features of Flexible Representation Use
In looking across the two episodes of Chad’s sense-making, we see similar
patterns of flexible representation use emerge:
1. Chad’s sense-making was highly reflexive in both cases. His representations
‘grew’ with his conceptual understandings of the two situations. In turn,
drawing and attending to the representations he had drawn helped develop his
conceptual understanding.
2. Chad generated highly coherent, coupled systems of representations. These
representations served different purposes, depending on emergent sense-
making or communication needs. Sense-making within one representation
often led to conceptual and representational consequences in another
representation.
3. Attention to new features or areas in his representations became generative
towards Chad’s sense-making, often through coordination across coupled
representations. Not only was noticing new features important for Chad to
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develop a deeper understanding, but acting on those features became
generative.
a. On the classical well problem; attention to the middle region in
his position representation, and straightening the middle
region, led to a deeper understanding of the probability
distribution of the bead.
b. On the slanted well problem; a new representational feature
(representational turning-point) in the slanted well
wavefunction emerged through the coordination of his infinite
well representation and the potential slant.
Overall, the above three points led to a reflexive form of sense-making in
which Chad repeatedly circled over his representations. As he circled over, Chad was
able to generate new meaning through the system he created and attending to new
areas within and across that system.
What’s dissimilar? The kinds of representational stepping stones Chad
introduces
On the two problems, Chad introduced various representations to serve as
‘stepping stones’ to reason towards his final products (wavefunctions). In the classical
well problem, Chad generated a ‘non-standard’ representation to reason with (plot of
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position versus time). I say it’s non-standard because it appears to be something Chad
constructs in the moment, based on the problem set-up. In any case, the representation
is some abstraction of the physical behavior of the bead. In other cases of students
reasoning on this problem, students use other representations of the bead to reason
about the bead’s behavior and its wavefunction. For example19, these may be
drawings of the bead-on-a-string, over which the student can simulate the bead’s
motion, then infer the wavefunction from the simulated behavior.
In contrast, on the slanted well problem, Chad recruited two iconic
representations of different quantum toy model situations. This pattern of using
quantum toy models as stepping stones in reasoning about the slanted well also holds.
Often, it is the infinite well. Others, like Chad, utilize the finite well too. This is
likely, in part, due to the conceptual and physical continuity between the infinite well
and the slanted well situations. However, it raises an important issue in supporting
flexible representation use in quantum. The issue is mainly a question of what types
of representations are available for students to sense-make with in quantum. It’s
unclear what a less-abstracted representation of the particle in the slanted well would
look like or whether it would be physically correct enough to be an appropriate tool to
reason with.
19 Quinn and Oliver in the previous chapter.
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Arguably, the most common quantum toy models (infinite well, finite well,
harmonic oscillator) are meant to be powerful examples of the application of the
Schrodinger Equation in different potentials. But clearly, they can also serve as
stepping stones towards understanding other situations. And so, this highlights the
importance of instruction providing students with opportunities to reason about
quantum behavior is a variety of situations.
Conclusions
In this chapter, I began with Greeno & Hall’s invocation for teachers to create
opportunities for students to use representations flexibly. Towards this purpose, I
described some design principles used in problem development and recounted two
episodes of student reasoning in the context of the designed problems that illustrates
flexible, reflexive use of representations. My goal is to expand upon the notion of
reflexive use of representation through my case studies of Chad, showing the coupled,
mutually influencing nature of Chad’s sense-making and his developing inscriptional
system. Through this case study, I hoped to begin challenging what counts as
proficiency in using representations. Reflexive use describes a pattern of sense-
making that may involve a back-and-forth between the traditionally characterized as
proficient actions of (re)-generating a representation to express one’s thinking and
appropriately reading-out from a representation. However, as I’ve have demonstrated,
inscriptions can provide more than a place to read-out information, but can be a site
for action.
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Designing disruptions to representational infrastructure became a lens to
critically examine the opportunities for innovation and learning available to students.
Thinking along these lines also provided a way to start thinking about what supports
are available to students in developing ‘disciplined inquiry’ when innovating and
inventing with representations. I found that the prompts encouraged innovation, with
interaction with the interviewer helping generate some coherence in Chad’s
reasoning. Bids for explanation sometimes treated Chad’s representations as piece-
wise, which may have encouraged Chad to sense-make around different features of
his representations. These bids often meant that Chad had to go back over
representations he had already drawn. Doing so helped initiate more ‘feed-back’
loops of reflexive sense-making and more coherent conceptual and representational
modeling of the two situations.
225
References
Ainsworth, S. (1999). The functions of multiple representations. Computers &
education, 33(2), 131-152.
Bolden, G. B. (2009). Implementing incipient actions: The discourse marker ‘so’ in
English conversation. Journal of Pragmatics, 41(5), 974-998.
Cataloglu, E., & Robinett, R. W. (2002). Testing the development of student
conceptual and visualization understanding in quantum mechanics through the
undergraduate career. American Journal of Physics, 70(3), 238-251.
Fredlund, T., Airey, J., & Linder, C. (2012). Exploring the role of physics
representations: an illustrative example from students sharing knowledge
about refraction. European Journal of Physics, 33(3), 657.
Gire, E., & Price, E. (2015). Structural features of algebraic quantum notations.
Physical Review Special Topics-Physics Education Research, 11(2), 020109.
Greeno, J. G., & Hall, R. P. (1997). Practicing representation. Phi Delta Kappan,
78(5), 361.
Hall, R. (1996). Representation as shared activity: Situated cognition and Dewey's
cartography of experience. The Journal of the Learning Sciences, 5(3), 209-
238.
Hall, R., Lehrer, R., Lucas, D., & Schauble, L. (2004, June). Of grids and jars: A
comparative analysis of representational infrastructure and learning
opportunities in middle school and professional science. In Proceedings of the
226
6th international conference on Learning sciences (pp. 238-245). International
Society of the Learning Sciences.
Hall, R., Stevens, R., & Torralba, T. (2002). Disrupting representational infrastructure
in conversations across disciplines. Mind, Culture, and Activity, 9(3), 179-
210.
Hammer, D., & Elby, A. (2003). Tapping epistemological resources for learning
physics. The Journal of the Learning Sciences, 12(1), 53-90.
Heckler, A. F. (2010). Some consequences of prompting novice physics students to
construct force diagrams. International Journal of Science Education, 32(14),
1829-1851.
Heller, P., & Hollabaugh, M. (1992). Teaching problem solving through cooperative
grouping. Part 2: Designing problems and structuring groups. American
Journal of Physics, 60(7), 637-644.
Heller, P., Keith, R., & Anderson, S. (1992). Teaching problem solving through
cooperative grouping. Part 1: Group versus individual problem solving.
American journal of physics, 60(7), 627-636.
Hutchins, E. (1995). Cognition in the Wild. MIT press.
Hutchins, E., & Klausen, T. (2000). Distributed Cognition in an Airline Cockpit.
Cognition and Communication at Work, 15–34.
Kirsh, D. (2010). Thinking with external representations. Ai & Society, 25(4), 441-
454.
227
Kohl, P. B., Rosengrant, D., & Finkelstein, N. D. (2007). Strongly and weakly
directed approaches to teaching multiple representation use in physics.
Physical Review Special Topics-Physics Education Research, 3(1), 010108.
Kuo, E., Hallinen, N. R., & Conlin, L. D. (2017). When procedures discourage
insight: epistemological consequences of prompting novice physics students
to construct force diagrams. International Journal of Science Education, 39(7),
814-839.
Latour, B. (1990). Drawing things together.
Lehrer, R., Schauble, L., Carpenter, S., & Penner, D. (2000). The inter-related
development of inscriptions and conceptual understanding. Symbolizing and
communicating in mathematics classrooms: Perspectives on discourse, tools,
and instructional design, 325-360.
Ma, J. Y. (2016). Designing disruptions for productive hybridity: The case of walking
scale geometry. Journal of the Learning Sciences, 25(3), 335-371.
McDermott, L. C., Rosenquist, M. L., & Van Zee, E. H. (1987). Student difficulties in
connecting graphs and physics: Examples from kinematics. American Journal
of Physics, 55(6), 503-513.
McKagan, S. B., & Wieman, C. E. (2006, February). Exploring student understanding
of energy through the quantum mechanics conceptual survey. In AIP
conference proceedings (Vol. 818, No. 1, pp. 65-68). AIP.
228
McKagan, S. B., Perkins, K. K., & Wieman, C. E. (2010). Design and validation of
the quantum mechanics conceptual survey. Physical Review Special Topics-
Physics Education Research, 6(2), 020121.
Nemirovsky, R. (1994). On ways of symbolizing: The case of Laura and the velocity
sign. The Journal of Mathematical Behavior, 13(4), 389-422.
Parnafes, O. (2007). What does “fast” mean? Understanding the physical world
through computational representations. The Journal of the Learning Sciences,
16(3), 415-450.
Reif, F., & Heller, J. I. (1982). Knowledge structure and problem solving in physics.
Educational psychologist, 17(2), 102-127.
Rumelhart, D. E., Smolensky, P., McClelland, J. L., & Hinton, G. E. (1986).
Schemata and sequential thought processes in PDP models, Parallel
distributed processing: explorations in the microstructure, vol. 2:
psychological and biological models. Chicago: Psychological and Biological
Models.
Schegloff, E. A. (2007). Sequence organization in interaction: Volume 1: A primer in
conversation analysis (Vol. 1). Cambridge University Press.
Schegloff, E. A., & Sacks, H. (1973). Opening up closings. Semiotica, 8(4), 289-327.
Schiffrin, D. (1988). Discourse markers (No. 5). Cambridge University Press.
Schoenfeld, A. H. (1985). Metacognitive and epistemological issues in mathematical
understanding. Teaching and learning mathematical problem solving: Multiple
research perspectives, 361-380.
229
Sidnell, J. (2011). Conversation analysis: An introduction (Vol. 45). John Wiley &
Sons.
Siegler, R. S., & Crowley, K. (1991). The microgenetic method: A direct means for
studying cognitive development. American psychologist, 46(6), 606.
Star, S. L., & Ruhleder, K. (1996). Steps toward an ecology of infrastructure: Design
and access for large information spaces. Information systems research, 7(1),
111-134.
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Chapter 5: Conclusions
The three studies in this dissertation focused mainly on studying student
thinking along various dimensions and providing insights to instructors based on
those studies. In this concluding chapter, I aim to expand on the ‘lessons learned’
from those studies. In doing so, I discuss the chapters individually.
Chapter 2: Tension in collaborative group-work
This chapter sought to model one way in interactional tension can function in
students’ conceptual reasoning. I analyzed three cases of students working in
collaborative to show tension in the groups became a driving mechanism in the
group’s taking an ‘escape hatch.’ In these escape hatches, students found creative
means to close a conversational topic while leaving some conceptual query
unresolved. In doing so, I showed the entanglement of various analytical dimensions
of interaction: social, epistemological, conceptual, and affective dimensions. The
entanglement of these dimensions of interaction, and that resolution multi-
dimensional tension through an escape hatch, leads to various implications for
research and instruction.
Implications for researchers
The three cases discussed all shows various ways in which cognitive, social,
emotional, epistemological dynamics of group interaction can feed into one another.
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Conflict within one layer often leads to conflict in another, occasionally resulting in
an escape hatch to relieve the multidimensional conflict and tension. This raises
question of whether there exists well-designed learning environments or interactional
patterns in which these dimensions are not so tightly coupled. I think it’s likely that
such design, or developed classroom norms, will never be able to fully decouple
emotion from conceptual sense-making. But there may be ways to make such
coupling less severe.
The issue of entanglement should provide some methodological insights to
researchers who may be interested in different types of conflict, as opposed to the
coupling of different types. For example, a researcher may be interested in
epistemological conflict. And so that researcher seeks out moments where students
are negotiating ‘what counts’ as knowledge in a particular context. Understanding
how and why students go about this negotiation should necessarily involve mapping
out tension and different types of conflict (conceptual, social, and epistemological) to
be able to fully contextualize the epistemological conflict and its resolution.
Instructional implications
In this work, we showed how the conversational closing in an escape hatch
can come in the forms of epistemological humor, exploiting tutorial wording, or
“agreeing to disagree,” (Lampert, Rittenhouse, & Crumbaugh 1996). These are just
particular instantiations of the broader interactional phenomenon of taking an escape
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hatch. Of course, they may come in other forms. The point of the work then is not to
direct instructor attention to these particular closings, to help instructors develop
practices of noticing that enable to see both; 1) entanglement in these different
analytical dimensions of interaction, 2) conversational closings that enable tension
and conflict resolution.
Understanding dimensions of interaction to be entangled has important
consequences for facilitating small group work. It shows that what facilitators are
responding to is not just in-the-moment sense-making, but the interactional history of
the group, as well. For example, a facilitator walking by a group may observe some
epistemological conflict being negotiated in a group. Maybe half the group is leaning
towards conceptual sense-making whereas the other half is arguing for more
quantitative approaches. Intervening immediately to help the group through their
negotiation may be unproductive, if the root cause of the tension experienced by the
group is not understood. For this reason, it becomes clear that some amount of
‘sampling’ needs to be done of group dynamics to better understand what exactly, as
a facilitator, you are actually responding to.
Students and emotions
A final point we advocated for is helping students notice conflict and
emotions in their interactions. This includes helping students see how interactional
tension and emotion function in their reasoning, more loading. The goal here being to
reconceptualize classrooms as a place where students can deal with emotionally-
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charged disagreements and find appropriate ways to find a resolution in that
disagreement. Resolution may be working towards a point of more clarity or being
explicit about why the discussion should be left and taken-up later on. Depending on
the context, both options are perfectly valid ways of dealing with disagreements.
Indeed, depending on the nature of the disagreement, the discussion may need to be
tabled many times over before the participants are able to find points of more clarity.
The hope is that this instructional orientation, towards understanding and managing
disagreement, would help students acquire tools to better engage with difficult issues,
both in and out of the classroom.
Chapters 3 and 4: flexible representation use
Both chapters 3 and 4 studied certain aspects of flexible representation use by
students. This line of work originated in seeing evidence of common student
difficulties in interview and focus group data. My noticing here was likely influenced
by the plethora of research on student difficulties in quantum. When I was able to
look past this, I began to notice the creativity and inventiveness of student’s
reasoning. And so this work sought to understand different ways that students use and
generate representations to develop new understanding and model new situations.
The works are quite complementary. However, I had a very specific reason for
separating, as I wanted to use chapter 2 to make a particular point to instructors.
Mainly, I try to encourage instructors to give students opportunities to sense-make
about more complex systems, even if they feel these students aren’t ready. Below, I’ll
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talk more about the inspiration for this chapter and then implications I have drawn
from this line of work.
Inspiration for Chapter 2: allowing students to move on to real-world
applications
Two summers ago, I was attending the annual Physics Education Research
Conference. The conference often has focused sessions and I was listening-in on
several researchers talk about their work in studying the teaching and learning of
quantum mechanics. Near the end of the session, when the panelists were taking
questions from the audience, a student made a request of the researchers (and
educators). He understood that a lot of new, fundamental knowledge is needed to
understand and learn quantum mechanics. But he was missing opportunities for
learning about real-world applications. In his words, “why the **** are these courses
not helping me understand how Pokemon-go works on my phone?”
Having a student push for seeing the application of what they’re learning is
really wonderful. It’s what instructors should hope for in their students. However, the
response from a senior researcher and educator was quite the opposite of wonderful;
“well, how far have you gotten [in quantum]?”
And so, this work aims to make the case to researchers and educators in
quantum mechanics, that students should be given opportunities to move on to more
complex applications. In doing so, I hope to elevate this one student’s voice and
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potentially others who are not privileged enough to be able to put themselves in
spaces to speak up for themselves.
Balancing different intuitions as an instructor
As mentioned above, this work is mainly responding to a pervasive
instructional notion that fundamentals should be mastered first. This notion of
fundamentals-first came up recently in a conversation with a well-respected physics
instructor who has written several textbooks, including one of quantum mechanics.
He expressed some tension emerging from differences in his instructor-intuition and
his intuition abstracted from being a learner. His instructor-intuition would encourage
him to make sure no student is left behind in understanding these fundamental
examples. But his intuition as a learner of physics leads him to realize that learning
itself is a non-linear process, and so mastering fundamentals continues to happen as
one learns about more complex systems. And so careful instruction should be a
balance of giving into these different intuitions, making sure not to prioritize one
consistently over the other.
Reconceptualizing toy models in QM: students should learn with and about toy
models
In showing different cases of students reasoning with toy models and their
iconic representations in new situations, I hoped to make a case for instructors to
reconceptualize what these toy models are for students. Instead of only being strong
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examples of the application of the Schrodinger Equation in simple potentials, they
can also be tools for learning with in making sense of new situations.
In this chapter, I illustrated three cases of students ‘learning with’ toy models
in order to show how the adaptation of the toy model could lead to a deeper
understanding of the toy model itself, as well as the new situation being modeled. In
particular, continuities that students saw between the two situations, and fleshing out
differences, helped direct student sense-making to particular areas in the toy models
and new situations. Most importantly, misunderstandings that students had about the
toy models, or sometimes their faltering at finding the words to describe the toy
model, did not prevent the students from using the toy model productively in
generating new meaning.
The main point for instructors here is that they should provide students’
opportunities to invent and adapt even if they think students aren’t ready. The process
of invention and adaptation may provide students opportunities to better learn
‘fundamentals’ that concern instructors. This perspective is complimented by
literature suggesting that even if early adaptation leads to incorrect results, i.e. the
process does end up providing the student with new meaning, early adaptation is still
valuable (Schwartz & Bransford, 1998; Schwartz & Martin, 2004). These studies
show that early invention can eventually lead to greater gains in student learning later
on.
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Chapter 4: Flexible Representation Use
I reproduce part of the discussion points to remind the reader of important
features of flexible representation use. Chad’s sense-making was highly reflexive in
both episodes I discussed. His representations ‘grew’ with his conceptual
understandings of the two situations. In turn, drawing and attending to the
representations he had drawn helped further develop his conceptual understanding.
1. Stepping Stone Representations: Chad generated highly coherent, coupled
systems of representations. These representations served different purposes,
depending on emergent sense-making or communication needs. Sense-making
within one representation often led to conceptual and representational
consequences in another representation.
2. Figural features: Attention to new features or areas in his representations
became generative towards Chad’s sense-making, often through coordination
across coupled representations. Not only was noticing new features important
for Chad to develop a deeper understanding, but acting on those features
became generative.
a. On the classical well problem; attention to the middle region in his position
representation, and straightening the middle region, led to a deeper understanding of
the probability distribution of the bead.
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b. On the slanted well problem; a new representational feature (representational
turning-point) in the slanted well wavefunction emerged through the coordination of
his infinite well representation and the potential slant.
Standard and non-standard Stepping Stone Representations
From analyses of the episodes with Chad and others not described in this
dissertation, it’s clear that ‘stepping-stone’ representations are often necessary for
student sense-making towards the final, ultimate representation that is typically the
goal in those instances. However, this raises two important instructional issues, 1)
what types of ‘stepping stones’ are available to students and 2) how do instructors
provide space for students to utilize stepping stones.
Comparing the two episodes with Chad, we see that different stepping stones
are available to his sense-making because of the physical context in which he is
reasoning. On the classical well problem, he is able to draw a graph of the bead’s
position versus time. Other students, such as Oliver and Quinn, drew pictures of the
bead-on-a-string and/or the velocity of the bead as representations to sense-make
with. In contrast, stepping stone representation on quantum problems tended to be
iconic representations associated with toy models. This thread, along with the work in
chapter 2, highlights the importance of toy models and their representations in
students’ reasoning.
Highlighting Figural Features
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In both chapters 3 and 4, I showed that student attention to and sense-making
around different figural features became integral in the generation of new meaning.
With the case of Chad in chapter 3, his sense-making was consistently coordinated
around particular figural features; the knots in the classical well problem and potential
walls on the slanted well problem. Focusing on these regions helped coordinate Chad
in developing a deeper understanding of regions outside of the focal features.
In the episodes with Quinn and Oliver, a similar pattern was present; their
sense-making focused around particular features or areas. In these cases, the infinite
well toy model helped direct their reasoning to the ‘same’ areas in the toy model and
bead-on-a-string. Unlike with Chad, these interviews proceed with the interviewer
highlighting new regions. Coordinating these new regions in their sense-making
provided a mechanism for Quinn and Oliver to develop deeper understandings of the
infinite well representation and the bead-on-a-string.
The differences in how students come to reason about new regions in their
representing and represented world's raises important instructional considerations. For
Chad, his reasoning about the bead-on-a-string naturally led him to consider
additional regions as his sense-making developed. However, it was important that
there was space in the classical well prompt for him to coordinate his reasoning
through a focus on the knots to begin with. I can imagine two scenarios that could
impede this focus in his reasoning. The first being a preceding question to this
prompt, as in on a tutorial, that positions all positions on the axis with equal
likelihood by asking something about what a measurement on the bead’s position
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would yield. Another way to impede this sense-making may be to overly simplify the
situation by suggesting the bounce at the ends happens instantaneously. And so, it
seems as if the conceptual space left in the prompt may help provide space for
students to sense-make in ways that are heavily coordinated by particular areas or
features.
While Chad’s sense-making led him to consider additional regions in his
representations as he came to a better understanding of both situations, it took
highlighting by the interviewer to get Quinn and Oliver to consider new regions. This
highlighting intervention became routinized across the collection of interviewers as
many students failed to consider off-center regions in their first modeling of the bead-
on-a-string. And so, it leads me to believe that this type of intervention could be
incorporated into a follow-up prompt.
Providing Opportunities for flexible representation use: thinking about
students’ access and opportunities
I turned to literature on designing disruptions to representational infrastructure
to think about designing for flexible representation use. In responding to disruptions
students need to be inventive and creative because they do not have to tools to
efficiently solve the problem at hand.
Thinking about disrupting representational infrastructure should lead
instructors to consider of more than just the conceptual content that students may
develop in response to a disruption. Disrupting representational infrastructure
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illuminates issues of access of opportunities for learning through considering what
opportunities are available to students through routines being developed in the course.
Getting a good grip on this can then help instructors think about ways to disrupt any
problematic norms. For example, students should have the opportunity to invent in
new situations because it can lead to the generation of new meaning and because it
may serve student desire to see applications of what they’re learning. Students should
also have access to instructional support, either through an individual and/or well-
designed curricular materials that support the students in adapting and inventing in a
disciplined way.
Takeaways from interviews
Some careful consideration is needed in taking these results, which come from
clinical contexts, and translating them into implications for instruction. Below, I’ll
briefly discuss a few of those differences that I project may be consequential for such
translation of findings:
1) While I was able to get to know students some over the course of
several interviews and/or focus groups, instructors have the benefit of
being able to get to know their students over a semester, or even
longer. When considering interactional norms and escape hatches, this
knowledge of the student can be a huge asset. Instructors can develop
a sense of what students work well together, what norms are
developing in the classroom, or what students might need more
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interventions than others. Such patterns can then help inform
instructors’ responsiveness to group-work.
2) Similar to the first point, instructors have the opportunity in
classrooms to spend time developing interactional and classroom
norms. These can involve shaping what types of thinking and
interactions are accepted in the classroom.
3) I believe it’s very likely that spaces, like classrooms, where students
are assessed, and where norms suggest that a student display canonical
knowledge, come at the expense of a learning space that supports
creativity and trying out non-canonical ideas. This means there may be
some tension associated with asking students to reason ’beyond’
norms that they’re used to. In interviews and focus-groups, I tried to
create spaces where students were encouraged to express their thinking
but finding the right answer (or not) was less valued. This may have
freed students to be more creative and flexible in their thinking,
because there was less risk of assessment.
4) Students themselves may see the interview or focus-group spaces as
very different from what’s accepted or normal in a classroom. This
may mean that the sense-making that emerged in the interview spaces
is not as accessible in the classroom. As mentioned above, this might
be remedied by working to set norms and expectations for what’s
expected in the class.
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References
Lampert, M., Rittenhouse, P., & Crumbaugh, C. (1996). Agreeing to disagree:
Developing sociable mathematical discourse. The handbook of education and
human development: New models of learning, teaching, and schooling, 731,
764.
Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and
instruction, 16(4), 475-5223.
Schwartz, D. L., & Martin, T. (2004). Inventing to prepare for future learning: The
hidden efficiency of encouraging original student production in statistics
instruction. Cognition and Instruction, 22(2), 129-184.
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Appendix 1
Transcript Conventions
The transcripts use the following protocols (Sacks, Schegloff, & Jefferson, 1974;
Jefferson, 2004). :
:: Elongated words or vowels
CAP Emphasized words are capitalized
[ Start of overlapping speech of first speaker is shown with open bracket
// Start of overlapping speech of second speaker
-- Turns that are cut off by other speakers or end abruptly are marked with a
hyphen
… Speaker turns that trail off are marked with an ellipsis
(()) actions other than speech, including gestures, are represented in italics and
surrounded by double parentheses
(?) Pieces of speech that are difficult to discern are preceded or replaced
(#) Length of a pause
Jefferson, G. (2004). Glossary of transcript symbols with an introduction. Pragmatics
and Beyond New Series, 125, 13-34.
Sacks, H., Schegloff, E. A., & Jefferson, G. (1974). A simplest systematics for the
organization of turn-taking for conversation. language, 696-735.
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Appendix 2
Tutorial on Doppler Cooling
Tutorial 8:
Doppler cooling
How does an ambulance sound speeding towards
you, as compared to sitting still or moving away?
Suppose we have an atom that can move along the
horizontal dimension, with laser light coming in from the right and from the left.
If the atom moves to the right, how do the wavelengths of each laser change, as seen
by the atom?
What about if the atom moves to the left?
Suppose the atom in #2 has an energy level structure, as shown to the right. The
arrow shows the energy of the photons produced by the lasers. This laser is “red-
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detuned” from the atomic transition because photons from the laser have a lower
frequency and energy, hence a longer or more-red wavelength, than the atomic
transition.
Draw an energy level diagram of the atom when it is moving to the right, showing the
photon energy of both lasers, as seen by the atom.
Does motion along the axis change the likelihood of excitation of the atom by
photons from one of the lasers? If so, which laser?
Consult an instructor before you proceed.
If the atom is moving to the right and absorbs a photon, how does the atom’s
momentum change, if at all? (Hint: A photon carries momentum.)
After absorbing the photon, the atom will then emit a photon, with an energy equal to
the
5P3/2 → 5S1/2 transition, in a random direction. Describe how the momentum of the
atom changes after many, many cycles of absorption from the right laser and random
emission.
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The diagram at right shows the energy absorbed and then emitted by the atom in one
cycle as seen by a motionless observer. So in each cycle, the atom emits more energy
than it absorbs. How can you reconcile this with the conservation of energy?
Let’s pull things together. Describe what happens to an atom
that is moving in one dimension between two red-detuned
laser beams that are shining in opposite directions (as shown
in the diagram in question #2).
Now let’s try to generalize to three dimensions. Instead of having an atom that is
confined to move in one dimension, it is now allowed to move in all three. How could
additional lasers be arranged and tuned so that the atom loses speed no matter which
way it’s moving?
If you had a large number of atoms in the system you described in #9, how does the
temperature of the atoms change as they undergo many cycles of absorption and
emission?
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Atoms in the system described in #9 are called an optical molasses. Why do you think
physicists chose that name?
Instructor Guide: Tutorial on Doppler Cooling
Instructor Guide 8:
Doppler cooling
This tutorial is designed to introduce students to the concept of laser cooling,
specifically Doppler cooling. By the end of the tutorial, the students should recognize
that counter-propagating, red-detuned laser beams may be used to create an optical
molasses, or a cloud of cool atoms. This tutorial can be used in conjunction with
Tutorial 9 (Zeeman Effect), where students see that a magnetic field can then be used
to provide spatial confinement of the cooled atoms. Taken together, the laser and
magnetic-field configurations described in Tutorials 8 and 9 form a Magneto-Optical
Trap (MOT). Doppler cooling can be used to cool atoms for atomic clocks. In
creating Bose-Einstein Condensates, scientists usually create a cloud of cooled atoms
with a MOT.
In the tutorial, some of the big ideas students will focus on are the following:
Doppler shift of sound and light
Atomic energy levels and transitions between levels
Laser-matter interactions
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Doppler Shift
1) Students should recognize that the sound waves from the ambulance are shifted to
a higher frequency when coming towards the observer, and a lower frequency when
moving away.
2) This is the same question, but on the Doppler shift for light. The lasers are blue-
shifted when the atom is moving towards them, and red-shifted when moving away.
Doppler Shifted Energy levels
3) There are many way to represent the energy levels, but many students choose to
draw diagrams similar to the one shown. Students should see that one laser (right)
gets shifted up in energy and the other laser gets shifted down.
4) Students should see that the laser that the atom is moving towards is shifted closer
to the atomic resonance.
Momentum Considerations
5) The photon’s momentum is absorbed by the atom. The atom gets a kick in the
direction of the laser’s propagation, essentially slowing the atom down in the
direction antiparallel to the laser’s propagation. The change in velocity that the atom
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receives is called the recoil velocity and can be calculated using the momentum of the
photon and mass of the atom. The recoil velocity is related to the minimum
temperature that is achievable through Doppler cooling.
6) Each cycle, the atom is slowed in its direction of motion through absorption and
then receives a kick in a random direction through emission. After many cycles, there
is a large slowing effect in the direction of motion, while the kicks that the atom gets
through reemission should average out to no net change.
Energy Considerations
7) One important consideration is that energy and energy conservation are frame-
dependent! This can be a difficult concept for students, as energy conservation is one
of the tools used most frequently in physics. In the lab frame, the atom does absorb a
photon of lower frequency than what it re-emits, leaving some small amount of
energy unaccounted for. This energy comes from the atom itself; kinetic energy is
being taken from the atom and given to the second photon.
8) The atom is slowed in its direction of motion, in both directions.
9) Three orthogonal pairs of counter-propagating, red-detuned beams.
10) The atoms are losing kinetic energy, so the temperature decreases.
11) Answers may vary. A molasses is something that moves slowly.
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Appendix 3
Tutorial on Zeeman Effect
Tutorial 9:
Zeeman effect
In this tutorial, we will investigate how a magnetic field and a pair of lasers might be used to
confine atoms in a small region of space.
1) Moving charges experience a force in a magnetic field. The potential
energy of the interaction is U = –μ·B, where μ is the magnetic moment of
the charge. It is proportional to the charge and points in the direction perpendicular to
the plane of motion.
a. If we have a charge that is forced to move in a circle, what happens to the
charge if we turn on a magnetic field that points in, for example, the positive
z-direction?
b. Does your answer above depend on what direction the magnetic moment
points in? Explain why or why not.
2) Atoms have angular momentum, which we can investigate by modeling the atom as
an electron orbiting a nucleus. If we apply a magnetic field in the positive z-direction,
what orientation of the atom has the lowest potential energy? (I.e., in what direction
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should the magnetic moment point?)
3) Thinking of the atom as a charge orbiting the nucleus is a classical model for a
quantum system. How accurately do you think this model describes the physical
situation?
4) Is there any value in using the classical model, even if we see that it breaks down
when we consider that an accelerated charge would radiate away energy? Explain
your reasoning.
5) For angular momentum of l =1, the magnetic quantum number m can take on values
of -1, 0 and 1. (The magnetic quantum number tells us the projection of the angular
momentum vector in a given direction, say z.) Which of these states (m = -1, 0, 1)
gives maximum, zero and minimum potential energy, given a magnetic field that
points in the positive z-direction?
6) In the absence of a magnetic field, the three levels would have the same energy. Fill
in the energy level diagram of the three magnetic levels splitting in the two cases
shown. On the left, the magnetic field is in the -z direction and on the right it is in the
+z direction. B0 is a constant.
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7) Suppose we
have the linear
magnetic field shown in the graph. Describe how the energy levels change as you
move away from the origin. Is it the same going in the +z direction as the -z
direction?
8) Time for a quick detour into polarization. (Polarization tells us which direction the
electric field of an electromagnetic wave points in.) Light can have many
polarizations, such as horizontal, vertical or circular. With circular polarization, the
E-field traces out a circle as the wave propagates. This
rotation can be clockwise or counterclockwise. We’ll call
these σ+ (or σ-), respectively. When an atom absorbs or emits
a σ+ photon, the electronic transition must satisfy Δm= +1.
Similarly, absorption or emission of a σ- photon allows a
transition with Δm = -1.
Which types of polarization (σ+, σ-, or none) can enable the
following transitions?
a) l =0, m=0 → l=1, m=+1
b) l=0, m=0 → l=1, m=0
c) l=0, m=0 → l=1, m=-1
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d) l=1, m=+1 → l=0, m=0
e) l=1, m=0 → l=0, m=0
f) l=1, m=-1 → l=0, m=0
9) Imagine we have the magnetic field described in question 7, one that varies linearly
with z-position. We can adapt our energy level diagram from problem 6 to the
diagram shown below. Label the magnetic m levels on the diagram.
10) Suppose we have a laser with an
energy level that is shown by the
dotted line on the diagram. The
atom will absorb a photon from the
laser if the energy of the photon is
close to the energy of an
appropriate transition of the atom.
If we have an atom in the ground state that is to the right of the origin, what kind of
polarization (σ+ or σ-) is it more likely to absorb?
What about a ground state atom to the left of the origin?
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Consult an instructor before you proceed.
11) A photon carries a momentum of ℏk, where k = 2πλ. If a stationary atom absorbs a
photon, where does the photon’s momentum go? Explain your reasoning.
12) If the photon in the previous question is coming in from the right, what is the velocity
of the atom (magnitude and direction) after the collision? This is called a recoil
velocity.
13) Now suppose we have the same linear magnetic field described in question 7, and
two lasers coming in from the right and left. The one from the left is σ+ while the one
from the right is σ-, and both carry photons with the energy depicted in the diagram
with question 10.
Describe what happens to ground state atoms at different positions along the z-axis.
(Note: in this situation, the atoms can only move along the z-axis.)
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14) When the atom absorbs a photon, it quickly re-emits another photon in a random
direction.
a) Imagine the atom is to the right of the origin. What is the effect of many, many cycles
of absorption and reemission?
b) What happens to an atom that is to the left of the origin after many cycles?
15) It’s possible that an atom that originally started to the right of the origin ends up
moving to the left past the origin after going through many cycles of reemission. Can
this atom move back towards the origin? Explain your reasoning.
16) Our goal was to confine atoms to a region near the origin. Have we accomplished
this? Why or why not?
17) Can we adapt this system to confine atoms in three dimensions? Explain your
trapping setup.
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Instructor Guide: Tutorial on Zeeman Effect
Instructor Guide 9:
Zeeman Effect
This tutorial is designed to introduce students to the Zeeman Effect. The students will
see that a magnetic field can then be used to provide spatial confinement of cooled
atoms. This tutorial should be used in conjunction with Tutorial 8 (Doppler cooling).
Taken together, the laser and magnetic-field configurations described in tutorials 8
and 9 form a Magneto-Optical Trap (MOT). Doppler cooling can be used to cool
atoms for atomic clocks. In creating Bose-Einstein Condensates, scientists usually
create a cloud of cooled atoms with a MOT.
In the tutorial, some of the big ideas students will focus on are the following:
Motion of charged particles in magnetic fields
Models of atomic orbits
Quantized angular momentum
Polarization
Momentum transfer
Motion in magnetic fields
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1a-b) This may be a bit of a trick question, as we don’t say what plane the atom
rotates in. The point of this question is that the students should see that this matters. If
students do not reach this point for whatever reason, they can move on to the next
question without the facilitator providing too much help, as 1b asks the students
directly whether the relative orientation matters. In any case, the rotating charge will
experience a torque, which will force the charge’s magnetic moment to align with the
magnetic field.
Students may struggle with the concept that a single charge can be modeled as
experiencing a torque. The facilitator may initiate the idea of an electron
attached to the end of a string, with the other end fixed in place. The electron
rotates quickly in a circle while in the presence of a weak magnetic field. Ask
the students to imagine how the plane of rotation would change in response to
this field; it should slowly tilt towards the the direction of the magnetic field.
2) The potential energy of the interaction is lowest when the magnetic moment and
magnetic field are aligned. This means the magnetic moment should point in the
positive z-direction.
To check for comprehension, a facilitator might ask what this magnetic
moment would imply for motion, e.g. asking the students what plane the
rotation lies in and what direction it goes. A magnetic moment in the positive
z-axis corresponds to counterclockwise motion in the x-y plane.
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3) The students’ answer to this question will not likely influence their work in the rest
of tutorial. This question provides a point for students to discuss some of the
interpretive issues in quantum mechanics. Some students may see the classical model
as a good description. Students may also look ahead and choose an answer based on
#4. The facilitator should choose whether they want to further discuss this point with
students.
4) The facilitator should use their judgment on how to approach this problem with
students. We would likely expect students to come up with the response that the
model (like the Bohr model) is useful in some ways, but maybe not all.
Quantized Angular Momentum
5) The states m = -1, 0, 1 would correspond to maximum, zero and minimum
energies, respectively. The state m=+1 is the state where the projection on the z-axis
is positive, so the angular momentum vector is above the x-y plane. Students may
lose track of the negative sign in the equation for potential energy, thinking that the
parallel configuration yields a maximum energy. Have the students check to make
sure their answers are consistent.
6) The states m = -1, 0, 1 would be maximum, zero and minimum energies on the
right side of the diagram. The states m = +1, 0, -1 would be maximum, zero and
minimum energies on the left side of the diagram.
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7) The states would be linear in energy with respect to the z-axis. The m=-1 state
would have a positive slope, the m=+1 state would have a negative slope and the m=0
state would remain horizontal. By #7, students should have come to see that energy
levels will shift in response to the magnetic field, but they may struggle with the idea
that the energy now depends on position along the z-axis. However, students will
need to fill a graph showing the linear tilts of the energy levels in #9. So if they are
unable to get the linear tilt, they will be given it in #9. There may be some confusion
here about what the axes of the graph represent; specifically, the vertical axis now
represents the magnitude of the magnetic field (rather than energy as in #6).
Polarization
8) Students may struggle with circular polarization. With circular polarization, the
electric field vector traces out a helix around the axis of propagation.
a. l =0, m=0 → l=1, m=+1
o Δm= +1, σ+
b. l=0, m=0 → l=1, m=0
o Δm= 0, none
c. l=0, m=0 → l=1, m=-1
o Δm= -1, σ-
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d. l=1, m=+1 → l=0, m=0
o Δm= -1, σ-
e. l=1, m=0 → l=0, m=0
o Δm= 0, none
f. l=1, m=-1 → l=0, m=0
o Δm= +1, σ+
9) The m=-1 state has the positive slope, the m=+1 state has the negative slope and
the m=0 state is horizontal.
10) The students should be looking for the energy levels that are shifted down, closer
to the wavelength of the laser. On the right side of the origin, the atom’s m=-1 energy
level is shifted closer to the energy of the laser. An atom here is unable to absorb σ-,
as σ- would can only drive a transition with Δm= -1, but m=-1 is the lowest magnetic
state available. Hence, an atom is more likely to absorb a σ+ photon and transition
from m=-1 to m=0. To the left of the origin, an atom is likely to absorb a σ-, and
transition from m=+1 to m=0.
Momentum Transfer
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11) The photon’s momentum is transferred to the particle, by conservation of
momentum. Throughout the tutorial, we draw both on the wave-like and particle-like
characteristics of light. For some students, it may be difficult to discuss light as an
electromagnetic wave with polarization and then consider what momentum a photon
can carry.
12) The momentum of the atom after absorption would be equal to the momentum of
the photon before. To find velocity, divide by mass, v = ℏkm. This velocity points in
the direction the photon was moving.
13) When atoms are to the right of the origin, the m=+1 state is closer to the laser’s
energy, making it more likely that the atom can absorb a σ- photon coming in from
the right side. On the left side, the m=-1 state is closer to the laser’s energy, which
makes it more likely that the atom can absorb a σ+ photon coming in from the left
side.
Atomic Confinement
14a-b) If the atom is to the right (left) of the origin, the atom gets many kicks towards
the origin. The kicks the atom receives from re-emission average out to zero.
263
15) Once the atom is to the left of the origin, it is closer in resonance to the laser
coming in from the left, making it more likely that the atom will get kicked back
towards the center.
16-17) In order to confine atoms in three dimension, we would have three orthogonal
pairs of counter-propagating lasers. Once students reach this point in the tutorial, have
them try to draw connections to the previous tutorial. A facilitator might ask whether
the temperature of the atom cloud has changed or how the speed of individual atoms
changes over time.
264
Appendix 4
Interview Protocol 1
1. Prompt:
a. Thanks for agreeing to do the interview! We’ll be doing the same kind
of thing we’ve been doing this semester, testing out some curriculum
materials that we might use in future quantum classes. So I just ask
that you think and work out loud. I’ll give you a few sheets of
problems but I just ask that you don’t look ahead because there’s
spoilers!
2. Quantitative/conceptual question: (Particle in a box) If you were to measure
the position of the particle at some point in time, what position(s) would you
expect to measure?
a. Follow-ups:
i. How did you get those values?
ii. If you were to repeat the measurement, would you get the same
value every time?
b. Goal: Contextual priming of particle in a box.
3. How confident do you feel in your answer to the position measurement
question?
a. If high confidence: How did you know that your answer was correct?
i. Do you generally feel confident in your answers?
ii. What makes you feel this way?
265
iii. How do you know when your answer to a problem is correct or
incorrect?
b. If low confidence: What made you feel that you didn’t have a correct
answer?
i. What makes you feel confident, or not, in your answers?
ii. How do you know when you have a correct or incorrect
answer?
c. Goal: This is hinting more at declarative knowledge, as we ask them
how they know in general when they’re right. However, I would call
this metacognitive priming, priming students to think about
metacognition.
4. With the particle in a box, if you were to measure the speed of the particle at
some point in time, what would you expect to measure and why?
a. How did you come to your answer?
b. Will you get the same measurement every time?
c. Does your answer make physical sense?
d. Goal: Contextual priming of particle in a box.
5. I gave this problem to a group of students and while they were solving, I heard
the following discussion.
a. Student 2: What would be the speed of the particle?
266
i. Student 1: Well it’s obviously getting at momentum. So the
expectation value of the momentum over the mass would give
us the speed, right?
ii. Student 1: Would the speed change? It’s just bouncing back
and forth.
iii. Student 2: Well, the speed wouldn’t change because the
potential is constant. But the velocity would have to change
direction at the walls.
b. As a student in the group, what might you say to your peers at this
point?
i. Goal: Awareness of the classical description of “just bouncing
back and forth.” This is assessing conditional knowledge, but
in context.
c. The conversation followed with:
i. Student 3: Well now it sounds like we’re switching from
quantum to classical explanations.
ii. Student 1: You’re right. So maybe we should stick to the
expectation value of momentum.
d. Can you comment on the contributions by each of these students to the
discussion?
i. I will have them comment on each student, if they don’t on
their own.
267
ii. Are their contributions different?
iii. Are there ways in which they are similar?
iv. How does it relate to your pretend role in the group?
6. Consider the hydrogen atom in the ground state. If you were to measure the
position of the electron, what might you measure?
a. How did you find that position?
b. Are there other positions where you might find the electron?
c. Is it equally likely that you can find it everywhere?
7. I recently overheard two students discussing what the speed of an electron in
a hydrogen atom might be.
a. Student 1 said, “picture the electron-proton as the moon-earth system,
we know that closer objects move faster. So a closer electron moves
faster.” To which the other student replied “that’s classical reasoning
for a quantum system.” Student 1 responds with “but the model is
classical mechanical and reproduces the correct energy levels, so it’s
okay to use.”
b. As a student in the group, how might you respond to your peers?
8. Here’s a tough one! Consider the particle in a box, but now slant the bottom of
the well. How might you go about finding the allowed energy eigenstates and
energies? (Need a diagram here!)
a. This problem is not solvable and will be used to assess student’s use of
planning and monitoring of possible solutions paths.
268
b. If the student jumps into solving:
i. What are doing?
ii. How does what you’re doing get you to the solution?
c. If the student appears stumped:
i. Can we take a few steps towards a solution?
9. I gave this problem to a group of students and heard the following:
a. When guessing what a possible energy eigenstate for the slanted
potential problem. A student makes the following statement. “Well,
the wavelength of the particle comes from the de Broglie wavelength,
which tells me that the wavelength is inversely proportional to
difference between the particle’s energy and the potential. This means
that in regions of low potential, where the difference between E and V
is large, the wavelength is small, and in regions of higher potential,
where the difference between E and V is small, the wavelength is
large. I can reason the amplitude of the wavefunction by picturing the
particle as a ball rolling up and down the hill. The amplitude of the
wavefunction is largest where the particle spends more time, this
would be at the top of the hill because the particle’s energy is all
potential energy here. So the amplitude should be large at the top of
the slant and small at the bottom.
b. What do you think about this student’s reasoning?
i. Does it appear to valid reasoning?
269
ii. Is there anything that makes you think twice about using this
type of reasoning?
c. This question is designed to test the conditional knowledge of classical
reasoning in terms of velocity. Can maybe get at the correspondence
principle as well.
Interview Protocol 2
This document contains the interview protocol for my next round of interviews. The
goal of the protocol follows the lines of analysis that we have been talking about in
the QM meetings recently, disruptions to representational infrastructure within
distributed cognition. Through the interview, I would like to first prompt canonical or
typical quantum inscriptions. Questions designed to elicit these representations are
labeled SW or SV, for Standard Written or Standard Verbal. Subsequent questions,
both verbal and written, are designed to disrupt the representational infrastructure
surrounding and including the inscription. These are labeled as verbal or written
questions (DV or DW).
Disruptions may change the physical form of the inscription or any of the layers of
the infrastructure below the representation. These layers include, but are not limited
to, tacit design choices, conventions of communication and conventions of use. While
it is not possible to determine a priori whether an interviewer move will be taken up
270
as a disruption by the student, it seems reasonable to attempt to introduce such
disruptions through conceptually difficult questions or re-framing typical prompts.
Both mechanisms for creating disruptions seem to fall within the conventions of use,
extending the representational infrastructure in conceptual and epistemological
dimensions.
My research questions seem to follow three interrelated threads: the canonical
representations that students present, distribution and redistribution of cognition in
the system, finding the mechanisms that initiated the redistribution.
1) What are the canonical representational systems that students are representing in
the interviews?
2) How is cognition distributed and redistributed throughout the system? or How are
changes in the representational infrastructure coupled to the emergent reasoning about
the physics? (I want to say that the second question does treat the system as the unit
of analysis, but sort of places an emphasis on the student's role.)
3) What causes the redistribution? or What causes disruptions to the representational
infrastructure?
Answering this first question, through my standard question types (written and
verbal), I will be able to show what representations students are bringing forth that
are historically and socially propagated forms. However, pursuing the second and
271
third threads will allow me to show that there's a lot more to the student's
understanding and reasoning than we might first anticipate by simply looking at the
canonical representations elicited through standard question. This would come in
conjunction with my analysis showing that the cognition is truly distributed across the
system, and to ask questions about the student's understanding and reasoning, we
must also answer how the reasoning/understanding is coupled to representation and
representational change.
PIAB
● SW: A quantum particle is in a region of potential defined as V(x)=∞ for x =
(-∞, 0) and (L,∞), and V(x) = 0 for x = (0,L). Sketch the wavefunction for the
first allowed state, or ground state, of the particle.
○ SV/DV: What are the values of the wavefunction at the edges?
○ SV/DV: What about the peak?
○ If students jump into heavy algebra, ask them how they would finish to
save time.
● SW: For the particle described in the previous problem, what are the first few
allowed energies and wavefunctions of the particle?
272
○ If the participant stacks the wavefunctions as in standard
representation:
■ SV/DV: Could you say more about why you plotted the
wavefunctions vertically?
■ Does the vertical stacking represent something physical?
■ SV/DV: What do the vertical and horizontal axes represent?
○ SV/DV: Is there more you can say using this drawing?
■ SV/DV: The system, the particle, energy, anything you want!
● SW: Suppose a particle in the potential described in problem 1 was excited to
the n=2 state. If you were to measure the position of the particle, what might
you measure?
○ DW: Suppose you performed an experiment to measure the position of
the particle in its n=2 state. You get a positive result at a specified
location. Where was the particle right before you measured it?
● DW: Suppose you had a classical particle in a physical situation analogous to
the quantum particle in the box. Consider a bead on a string, and the string is
knotted at x = 0 and x = L so that the bead is confined between 0 to L, but can
move smoothly and freely between these bounds. The bead has some energy,
E, and can bounce elastically at the knotted ends. What is the wavefunction of
this classical particle?
○ If they are unsure about applying the wavefunction idea to a classical
particle
273
■ DV: Is there anything about the wavefunction idea that you can
bring in to understand the motion of the classical particle? Is
there anything about that kind of thinking that you can bring in
here?
○ DW: Compare this wavefunction to that of the quantum particle in the
ground state and in a very highly excited state.
● DW: Suppose a friend from your quantum class missed the lecture on the
particle in a box. How would you explain it to your friend?
○ If unclear on “it”
■ DV: Essentially what you put for problem 1
● DW: How might you explain your response to problem 1 to a friend who’s an
English major?
Tunneling through a barrier (if time, keep separate)
● SW: A free quantum particle with energy E coming in from the left
encounters a potential barrier of the form, V(x) = 0 from (-∞, 0) and (a, ∞).
From (0,a), the potential is V(x)= V0. (consider E<Vo)
○ SW: Sketch what the wavefunction of the particle will look like.
○ SW: If you were to measure the position of the particle, where are you
most likely to find the particle?
○ DV: Can you describe what the particle is doing?
○ DV: Can you explain the relative heights of the different pieces of the
wavefunction?
274
○ DV: Can you describe the probability of finding the particle inside the
barrier?
● DW: What would the wavefunction of the particle look like if it had an energy
above that of the potential barrier?
○ DV: Can you describe what the particle is doing?
275
Bibliography
Baily, C., & Finkelstein, N. D. (2009). Development of quantum perspectives in
modern physics. Physical Review Special Topics-Physics Education Research,
5(1), 010106.
Baily, C., & Finkelstein, N. D. (2010). Teaching and understanding of quantum
interpretations in modern physics courses. Physical Review Special Topics-
Physics Education Research, 6(1), 010101.
Bao, L. (1999). Dynamics of student modeling: A theory, algorithms, and application
to quantum mechanics.
Bao, L., & Redish, E. F. (2002). Understanding probabilistic interpretations of
physical systems: A prerequisite to learning quantum physics. American Journal
of Physics, 70(3), 210-217.
Barron, B. (2000). Achieving coordination in collaborative problem-solving groups.
The Journal of the Learning Sciences, 9(4), 403-436.
Barron, B. (2003). When smart groups fail. The journal of the learning sciences,
12(3), 307-359.
Berland, L. K. and Reiser, B. J. (2011), Classroom communities' adaptations of the
practice of scientific argumentation. Sci. Ed., 95: 191–216.
Blumenfeld, P. C., Marx, R. W., Soloway, E., & Krajcik, J. (1996). Learning with
peers: From small group cooperation to collaborative communities. Educational
researcher, 25(8), 37-39.
276
Bolden, G. B. (2009). Implementing incipient actions: The discourse marker ‘so’ in
English conversation. Journal of Pragmatics, 41(5), 974-998.
Bransford, J. D., & Schwartz, D. L. (1999). Chapter 3: Rethinking transfer: A simple
proposal with multiple implications. Review of research in education, 24(1), 61-
100.
Bricker, L. A., & Bell, P. (2008). Conceptualizations of argumentation from science
studies and the learning sciences and their implications for the practices of science
education. Science Education, 92(3), 473–498.
Brizuela, B. M. (2004). Mathematical development in young children: Exploring
notations. Teachers College Press.
Brookes, D. T., & Etkina, E. (2007). Using conceptual metaphor and functional
grammar to explore how language used in physics affects student learning.
Physical Review Special Topics-Physics Education Research, 3(1), 010105.
Brown, A. L., & Palincsar, A. S. (1989). Guided, cooperative learning and individual
knowledge acquisition. Knowing, learning, and instruction: Essays in honor of
Robert Glaser, 393-451.
Buchs, C., Butera, F., Mugny, G., & Darnon, C. (2004). Conflict elaboration and
cognitive outcomes. Theory into Practice, 43(1), 23-30.
Carraher, D., Schliemann, A. D., & Brizuela, B. M. (2000). Early algebra, early
arithmetic: Treating operations as functions. In Presentado en the Twenty–second
annual meeting of the North American Chapter of the International Group for the
Psychology of Mathematics Education, Tucson, Arizona.
277
Carraher, D., Schliemann, A., & Brizuela, B. (1999). Bringing out the algebraic
character of arithmetic. In AERA Meeting (40 pp.).
Cataloglu, E., & Robinett, R. W. (2002). Testing the development of student
conceptual and visualization understanding in quantum mechanics through the
undergraduate career. American Journal of Physics, 70(3), 238-251.
Collins, A., Brown, J.S., & Newman, S.E. (1989). Cognitive apprenticeship:
Teaching the crafts of reading, writing, and mathematics. In L. Resnick (Ed.),
Knowledge, learning and instruction: Essays in honor of Robert Glaser (453-494).
Hillsdale, NJ: Erlbaum.
Conlin, L. D. (2012). Building shared understandings in introductory physics tutorials
through risk, repair, conflict & comedy.
Csikszentmihalyi, M., & Sawyer, K. (2014). Shifting the focus from individual to
organizational creativity. In The Systems Model of Creativity (pp. 67-71).
Springer Netherlands.
Daoud, M., & Hussin, V. (2002). General sets of coherent states and the Jaynes–
Cummings model. Journal of Physics A: Mathematical and General, 35(34), 7381.
De Dreu, C. K., & Weingart, L. R. (2003). Task versus relationship conflict, team
performance, and team member satisfaction: a meta-analysis. Journal of applied
Psychology, 88(4), 741.
De Dreu, C. K., & West, M. A. (2001). Minority dissent and team innovation: the
importance of participation in decision making. Journal of applied Psychology,
86(6), 1191.
278
Derry, S. J., Pea, R. D., Barron, B., Engle, R. A., Erickson, F., Goldman, R. & Sherin,
B. L. (2010). Conducting video research in the learning sciences: Guidance on
selection, analysis, technology, and ethics. The Journal of the Learning Sciences,
19(1), 3-53.
Derry, S. J., Pea, R. D., Barron, B., Engle, R. A., Erickson, F., Goldman, R.,& Sherin,
B. L. (2010). Conducting video research in the learning sciences: Guidance on
selection, analysis, technology, and ethics. The Journal of the Learning Sciences,
19(1), 3-53.
diSessa, A. A. (1991). Epistemological micromodels: The case of coordination and
quantities. Psychologie génétique et sciences cognitives, 169-194.
diSessa, A. A. (1993). Toward an epistemology of physics. Cognition and instruction,
10(2-3), 105-225.
diSessa, A. A., Levin, M., & Brown, N. (edited volume, to appear in 2015).
Knowledge and interaction: A synthetic agenda for the learning sciences. New
York, NY: Routledge.
Domert, D., Linder, C., & Ingerman, Å. (2004). Probability as a conceptual hurdle to
understanding one-dimensional quantum scattering and tunnellingtunneling.
European Journal of Physics, 26(1), 47.
Dósa, K., & Russ, R. (2016). Beyond Correctness: Using Qualitative Methods to
Uncover Nuances of Student Learning in Undergraduate STEM Education.
Journal of College Science Teaching, 46(2), 70.
279
Dunbar, K. (1997). How scientists think: On-line creativity and conceptual change in
science. Creative thought: An investigation of conceptual structures and
processes, 4.
Einstein, A. (1936). Physics and reality. Journal of the Franklin Institute, 221(3), 349-
382.
Emigh, P. J., Passante, G., & Shaffer, P. S. (2015). Student understanding of time
dependence in quantum mechanics. Physical Review Special Topics-Physics
Education Research, 11(2), 020112.
Erickson, F. (2006). Definition and analysis of data from videotape: Some research
procedures and their rationales. Handbook of complementary methods in
education research, 3, 177-192.
Flavell, J. H. (1979). Metacognition and cognitive monitoring: A new area of
cognitive–developmental inquiry. American psychologist, 34(10), 906.
Fredlund, T., Airey, J., & Linder, C. (2012). Exploring the role of physics
representations: an illustrative example from students sharing knowledge about
refraction. European Journal of Physics, 33(3), 657.
Gibson, J. J., & Gibson, E. J. (1955). Perceptual Learning: Differentiation or
enrichment. Psychological Review, 62, 32-51.
Gire, E., & Price, E. (2015). Structural features of algebraic quantum notations.
Physical Review Special Topics-Physics Education Research, 11(2), 020109.
Goffman, E. (1955). On face-work: An analysis of ritual elements in social
interaction. Psychiatry, 18(3), 213-231.
280
Goodwin, C. (1994). Professional vision. American anthropologist, 96(3), 606-633.
Goodwin, C. (2007). Participation, stance and affect in the organization of activities.
Discourse & Society, 18(1), 53-73.
Goodwin, M. H., Goodwin, C., & Yaeger-Dror, M. (2002). Multi-modality in girls’
game disputes. Journal of pragmatics, 34(10), 1621-1649.
Greeno, J. G., & Hall, R. P. (1997). Practicing representation. Phi Delta Kappan,
78(5), 361.
Greer, L. L., Jehn, K. A., & Mannix, E. A. (2008). Conflict transformation a
longitudinal investigation of the relationships between different types of
intragroup conflict and the moderating role of conflict resolution. Small Group
Research, 39(3), 278-302.
Griffiths, D. J. (2016). Introduction to quantum mechanics. Cambridge University
Press.
Hall, R. (1996). Representation as shared activity: Situated cognition and Dewey's
cartography of experience. The Journal of the Learning Sciences, 5(3), 209-238.
Hall, R., Lehrer, R., Lucas, D., & Schauble, L. (2004, June). Of grids and jars: A
comparative analysis of representational infrastructure and learning opportunities
in middle school and professional science. In Proceedings of the 6th international
conference on Learning sciences (pp. 238-245). International Society of the
Learning Sciences.
Hall, R., Stevens, R., & Torralba, T. (2002). Disrupting representational infrastructure
in conversations across disciplines. Mind, Culture, and Activity, 9(3), 179-210.
281
Hammer, D. (2000). Student resources for learning introductory physics. American
Journal of Physics, 68(S1), S52-S59.
Hammer, D., & Elby, A. (2003). Tapping epistemological resources for learning
physics. The Journal of the Learning Sciences, 12(1), 53-90.
Hammer, D., Elby, A., Scherr, R. E., & Redish, E. F. (2005). Resources, framing, and
transfer. Transfer of learning from a modern multidisciplinary perspective, 89.
Hatano, G. Inagaki, K. (1986). Two courses of expertise. Child development and
education in Japan. NY: WH Freeman.
Hatano, G., & Oura, Y. (2003). Commentary: Reconceptualizing school learning
using insight from expertise research. Educational researcher, 32(8), 26-29.
Heckler, A. F. (2010). Some consequences of prompting novice physics students to
construct force diagrams. International Journal of Science Education, 32(14),
1829-1851.
Heller, P., & Hollabaugh, M. (1992). Teaching problem solving through cooperative
grouping. Part 2: Designing problems and structuring groups. American Journal
of Physics, 60(7), 637-644.
Heller, P., Keith, R., & Anderson, S. (1992). Teaching problem solving through
cooperative grouping (Part 1): Group versus individual problem solving.
American Journal of Physics, 60(7), 627-636.
Heller, P., Keith, R., & Anderson, S. (1992). Teaching problem solving through
cooperative grouping. Part 1: Group versus individual problem solving. American
journal of physics, 60(7), 627-636.
282
Hestenes, D., Wells, M., & Swackhamer, G. (1992). Force concept inventory. The
physics teacher, 30(3), 141-158.
Hutchins, E. (1995). Cognition in the Wild. MIT press.
Hutchins, E., & Klausen, T. (2000). Distributed Cognition in an Airline Cockpit.
Cognition and Communication at Work, 15–34.
https://doi.org/http://dx.doi.org/10.1017/CBO9781139174077.002
Intuitive Quantum Physics, (http://umaine.edu/per/projects/iqp/)
Jefferson, G. (2004). A note on laughter in ‘male–female’ interaction. Discourse
Studies, 6(1), 117-133.
Jehn, K. A. (1995). A multimethod examination of the benefits and detriments of
intragroup conflict. Administrative science quarterly, 256-282.
Jehn, K. A. (1997). A qualitative analysis of conflict types and dimensions in
organizational groups. Administrative science quarterly, 530-557.
Jehn, K. A., & Mannix, E. A. (2001). The dynamic nature of conflict: A longitudinal
study of intragroup conflict and group performance. Academy of management
journal, 44(2), 238-251.
John-Steiner, V. (2000). Creative collaboration. Oxford University Press.
Johnson, D. W., & Johnson, R. T. (1979). Conflict in the classroom: Controversy and
learning. Review of Educational Research, 49(1), 51-69.
Johnston, I. D., Crawford, K., & Fletcher, P. R. (1998). Student difficulties in
learning quantum mechanics. International Journal of Science Education, 20(4),
427-446.
283
Jordan, B., & Henderson, A. (1995). Interaction analysis: Foundations and practice.
The journal of the learning sciences, 4(1), 39-103.
Kalkanis, G., Hadzidaki, P., & Stavrou, D. (2003). An instructional model for a
radical conceptual change towards quantum mechanics concepts. Science
education, 87(2), 257-280.
Ke, J. L., Monk, M., & Duschl, R. (2005). Learning Introductory Quantum Physics:
Sensori‐motor experiences and mental models. International Journal of Science
Education, 27(13), 1571-1594.
Kelly, G. J., Druker, S., & Chen, C. (1998). Students’ reasoning about electricity:
Combining performance assessments with argumentation analysis. International
Journal of Science Education, 20(7), 849-871.
Kelly, G. J., McDonald, S., & Wickman, P. O. (2012). Science learning and
epistemology. In Second international handbook of science education (pp. 281-
291). Springer Netherlands.
Kirsh, D. (2010). Thinking with external representations. Ai & Society, 25(4), 441-
454.
Kohl, P. B., Rosengrant, D., & Finkelstein, N. D. (2007). Strongly and weakly
directed approaches to teaching multiple representation use in physics. Physical
Review Special Topics-Physics Education Research, 3(1), 010108.
Kuo, E., Hallinen, N. R., & Conlin, L. D. (2017). When procedures discourage
insight: epistemological consequences of prompting novice physics students to
284
construct force diagrams. International Journal of Science Education, 39(7), 814-
839.
Kutnick, P. J. (1990). A social critique of cognitively based science curricula. Science
Education, 74(1), 87–94.
Lampert, M., Rittenhouse, P., & Crumbaugh, C. (1996). Agreeing to disagree:
Developing sociable mathematical discourse. The handbook of education and
human development: New models of learning, teaching, and schooling, 731, 764.
Latour, B. (1990). Drawing things together.
Lave, J., & Wenger, E. (1998). Communities of practice: Learning, meaning, and
identity.
Lawson, C. A. (1956). An analysis of the process by which a group selects or rejects
ideas or beliefs. Science Education, 40(4), 245–253.
Lehrer, R., Schauble, L., Carpenter, S., & Penner, D. (2000). The inter-related
development of inscriptions and conceptual understanding. Symbolizing and
communicating in mathematics classrooms: Perspectives on discourse, tools, and
instructional design, 325-360.
Lemke, J. L. (1990). Talking science: Language, learning, and values. Ablex
Publishing Corporation, 355 Chestnut Street, Norwood, NJ 07648 (hardback:
ISBN-0-89391-565-3; paperback: ISBN-0-89391-566-1).
Liboff, R. L. (2003). Introductory quantum mechanics. Pearson Education India.
Lin, S. Y., & Singh, C. (2009). Categorization of quantum mechanics problems by
professors and students. European Journal of Physics, 31(1), 57.
285
Lovelace, K., Shapiro, D. L., & Weingart, L. R. (2001). Maximizing cross-functional
new product teams' innovativeness and constraint adherence: A conflict
communications perspective. Academy of management journal, 44(4), 779-793.
Lytra, V. (2009). Constructing academic hierarchies: Teasing and identity work
among peers at school. Pragmatics, 19(3), 449-66.
Ma, J. Y. (2016). Designing disruptions for productive hybridity: The case of walking
scale geometry. Journal of the Learning Sciences, 25(3), 335-371.
Mannila, K., Koponen, I. T., & Niskanen, J. A. (2001). Building a picture of students'
conceptions of wave-and particle-like properties of quantum entities. European
journal of physics, 23(1), 45.
Marshman, E., & Singh, C. (2015). Framework for understanding the patterns of
student difficulties in quantum mechanics. Physical Review Special Topics-
Physics Education Research, 11(2), 020119.
McDermott, L. C. (2001). Oersted medal lecture 2001:“Physics Education
Research—the key to student learning”. American Journal of Physics, 69(11),
1127-1137.
McDermott, L. C., Rosenquist, M. L., & Van Zee, E. H. (1987). Student difficulties in
connecting graphs and physics: Examples from kinematics. American Journal of
Physics, 55(6), 503-513.
McKagan, S. B., & Wieman, C. E. (2006, February). Exploring student understanding
of energy through the quantum mechanics conceptual survey. In AIP conference
proceedings (Vol. 818, No. 1, pp. 65-68). AIP.
286
McKagan, S. B., Perkins, K. K., & Wieman, C. E. (2008). Deeper look at student
learning of quantum mechanics: The case of tunneling. Physical Review Special
Topics-Physics Education Research, 4(2), 020103.
McKagan, S. B., Perkins, K. K., & Wieman, C. E. (2010). Design and validation of
the quantum mechanics conceptual survey. Physical Review Special Topics-
Physics Education Research, 6(2), 020121.
Michaels, S., O’Connor, C., & Resnick, L. B. (2008). Deliberative discourse idealized
and realized: Accountable talk in the classroom and in civic life. Studies in
philosophy and education, 27(4), 283-297.
Morgan, J. T., Wittmann, M. C., & Thompson, J. R. (2003). Student understanding of
tunneling in quantum mechanics: examining survey results.
Mortimer, E. F., & Machado, A. H. (2000). Anomalies and conflicts in classroom
discourse. Science Education, 84(4), 429–444.
Müller, R., & Wiesner, H. (2002). Teaching quantum mechanics on an introductory
level. American Journal of physics, 70(3), 200-209.
Nanda, K. K., Kruis, F. E., Fissan, H., & Behera, S. N. (2004). Effective mass
approximation for two extreme semiconductors: Band gap of PbS and CuBr
nanoparticles. Journal of applied physics, 95(9), 5035-5041.
Nemirovsky, R. (1994). On ways of symbolizing: The case of Laura and the velocity
sign. The Journal of Mathematical Behavior, 13(4), 389-422.
Olsen, R. V. (2002). Introducing quantum mechanics in the upper secondary school: a
study in Norway. International Journal of Science Education, 24(6), 565-574.
287
Ono, Y., Zimmerman, N. M., Yamazaki, K., & Takahashi, Y. (2003). Turnstile
Operation Using a Silicon Dual-Gate Single-Electron Transistor. Japanese Journal
of Applied Physics, Part 2: Letters, 42(10 A).
Parnafes, O. (2007). What does “fast” mean? Understanding the physical world
through computational representations. The Journal of the Learning Sciences,
16(3), 415-450.
Parnafes, O., & Disessa, A. A. (2013). Microgenetic learning analysis: A
methodology for studying knowledge in transition. Human Development, 56(1),
5-37.
Passante, G., Emigh, P. J., & Shaffer, P. S. (2015). Examining student ideas about
energy measurements on quantum states across undergraduate and graduate
levels. Physical Review Special Topics-Physics Education Research, 11(2),
020111.
Petri, J., & Niedderer*, H. (1998). A learning pathway in high‐school level quantum
atomic physics. International Journal of Science Education, 20(9), 1075-1088.
QuILT, (https://www.physport.org/examples/quilts/index.cfm).
Rebello, N. S. (2009, November). Can we assess efficiency and innovation in
transfer?. In AIP Conference Proceedings(Vol. 1179, No. 1, pp. 241-244). AIP.
Reif, F., & Heller, J. I. (1982). Knowledge structure and problem solving in physics.
Educational psychologist, 17(2), 102-127.
288
Roschelle, J., & Teasley, S. D. (1995). The construction of shared knowledge in
collaborative problem solving. In Computer supported collaborative learning (pp.
69-97). Springer Berlin Heidelberg.
Rosebery, A. S., & Puttick, G. M. (1998). Teacher professional development as
situated sense‐making: A case study in science education. Science Education,
82(6), 649-677.
Rosebery, A. S., Ogonowski, M., DiSchino, M., & Warren, B. (2010). “The coat traps
all your body heat”: Heterogeneity as fundamental to learning. The Journal of the
Learning Sciences, 19(3), 322-357.
Rumelhart, D. E., Smolensky, P., McClelland, J. L., & Hinton, G. E. (1986).
Schemata and sequential thought processes in PDP models, Parallel distributed
processing: explorations in the microstructure, vol. 2: psychological and
biological models. Chicago: Psychological and Biological Models.
Sacks, H., Schegloff, E. A., & Jefferson, G. (1974). A simplest systematics for the
organization of turn-taking for conversation. language, 696-735.
Sadaghiani, H. R. (2005). Conceptual and mathematical barriers to students learning
quantum mechanics (Doctoral dissertation, The Ohio State University).
Sadaghiani, H. R., & Pollock, S. J. (2015). Quantum mechanics concept assessment:
Development and validation study. Physical Review Special Topics-Physics
Education Research, 11(1), 010110.
289
Sadaghiani, H., & Bao, L. (2006, February). Student difficulties in understanding
probability in quantum mechanics. In AIP Conference Proceedings (Vol. 818, No.
1, pp. 61-64). AIP.
Savinainen, A., & Scott, P. (2002). The Force Concept Inventory: a tool for
monitoring student learning. Physics Education, 37(1), 45.
Schegloff, E. A. (2007). Sequence organization in interaction: Volume 1: A primer in
conversation analysis (Vol. 1). Cambridge University Press.
Schegloff, E. A., & Sacks, H. (1973). Opening up closings. Semiotica, 8(4), 289-327.
Scherr, R. E., & Hammer, D. (2009). Student behavior and epistemological framing:
Examples from collaborative active-learning activities in physics. Cognition and
Instruction, 27(2), 147-174.
Schiffrin, D. (1988). Discourse markers (No. 5). Cambridge University Press.
Schliemann, A. D. (2013). Bringing out the algebraic character of arithmetic: From
children's ideas to classroom practice. Routledge.
Schoenfeld, A. H. (1985). Metacognitive and epistemological issues in mathematical
understanding. Teaching and learning mathematical problem solving: Multiple
research perspectives, 361-380.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving,
metacognition, and sense making in mathematics. Handbook of research on
mathematics teaching and learning, 334-370.
Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and
instruction, 16(4), 475-5223.
290
Schwartz, D. L., & Martin, T. (2004). Inventing to prepare for future learning: The
hidden efficiency of encouraging original student production in statistics
instruction. Cognition and Instruction, 22(2), 129-184.
Schwartz, D. L., Bransford, J. D., & Sears, D. (2005). Efficiency and innovation in
transfer. Transfer of learning from a modern multidisciplinary perspective, 1-51.
Serway, R. A., Moses, C. J., & Moyer, C. A. (2004). Modern physics. Cengage
Learning.
Shah, P. P., & Jehn, K. A. (1993). Do friends perform better than acquaintances? The
interaction of friendship, conflict, and task. Group decision and negotiation, 2(2),
149-165.
Shirouzu, H., Miyake, N., & Masukawa, H. (2002). Cognitively active externalization
for situated reflection. Cognitive science, 26(4), 469-501.
Sidnell, J. (2011). Conversation analysis: An introduction (Vol. 45). John Wiley &
Sons.
Siegler, R. S., & Crowley, K. (1991). The microgenetic method: A direct means for
studying cognitive development. American psychologist, 46(6), 606.
Simons, T. L., & Peterson, R. S. (2000). Task conflict and relationship conflict in top
management teams: the pivotal role of intragroup trust. Journal of applied
psychology, 85(1), 102.
Singh, C. (2001). Student understanding of quantum mechanics. American Journal of
Physics, 69(8), 885-895.
291
Singh, C. (2008). Interactive learning tutorials on quantum mechanics. American
Journal of Physics, 76(4), 400-405.
Singh, C., & Marshman, E. (2015). Review of student difficulties in upper-level
quantum mechanics. Physical Review Special Topics-Physics Education
Research, 11(2), 020117.
Smith III, J. P., Disessa, A. A., & Roschelle, J. (1994). Misconceptions reconceived:
A constructivist analysis of knowledge in transition. The journal of the learning
sciences, 3(2), 115-163.
Star, S. L., & Griesemer, J. R. (1989). Institutional ecology, translations' and
boundary objects: Amateurs and professionals in Berkeley's Museum of
Vertebrate Zoology, 1907-39. Social studies of science, 19(3), 387-420.
Star, S. L., & Ruhleder, K. (1996). Steps toward an ecology of infrastructure: Design
and access for large information spaces. Information systems research, 7(1), 111-
134.
Steinberg, R. N., Oberem, G. E., & McDermott, L. C. (1996). Development of a
computer‐based tutorial on the photoelectric effect. American Journal of Physics,
64(11), 1370-1379.
Steinberg, R., Wittmann, M. C., Bao, L., & Redish, E. F. (1999). The influence of
student understanding of classical physics when learning quantum mechanics.
Research on teaching and learning quantum mechanics, 41-44.
292
Stevens, R., & Hall, R. (1998). Disciplined perception: Learning to see in
technoscience. Talking mathematics in school: Studies of teaching and learning,
107-149.
Stivers, T., & Sidnell, J. (2005). Introduction: multimodal interaction. Semiotica,
2005(156), 1-20.
Sullivan, F. R., & Wilson, N. C. (2015). Playful talk: Negotiating opportunities to
learn in collaborative groups. Journal of the Learning Sciences, 24(1), 5-52.
Thornton, R. K., & Sokoloff, D. R. (1998). Assessing student learning of Newton’s
laws: The force and motion conceptual evaluation and the evaluation of active
learning laboratory and lecture curricula. American Journal of Physics, 66(4),
338-352.
Tutorials in Physics: Quantum Mechanics,
(https://depts.washington.edu/uwpeg/tutorials-QM)
Van den Bossche, P., Gijselaers, W., Segers, M., Woltjer, G., & Kirschner, P. (2011).
Team learning: building shared mental models. Instructional Science, 39(3), 283-
301.
Warren, B., Ogonowski, M., & Pothier, S. (2005). “Everyday” and “scientific”: Re-
thinking dichotomies in modes of thinking in science learning. Everyday Matters
in Science and Mathematics: Studies of Complex Classroom Events, 119–148.
Wittmann, M. C., Morgan, J. T., & Bao, L. (2005). Addressing student models of
energy loss in quantum tunnelling. European Journal of Physics, 26(6), 939.
293
Wittmann, M. C., Steinberg, R. N., & Redish, E. F. (2002). Investigating student
understanding of quantum physics: Spontaneous models of conductivity.
American Journal of Physics, 70(3), 218-226.
Zhu, G., & Singh, C. (2012). Improving students’ understanding of quantum
measurement. I. Investigation of difficulties. Physical Review Special Topics-
Physics Education Research, 8(1), 010117.
Zhu, G., & Singh, C. (2012). Improving students’ understanding of quantum
measurement. II. Development of research-based learning tools. Physical Review
Special Topics-Physics Education Research, 8(1), 010118.